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A rectangular array
+
A rectangular array  
 +
$$\begin{pmatrix}
 +
a_{11}&\dots&a_{1n}\\
 +
\dots&\dots&\dots\\
 +
a_{m1}&\dots&a_{mn}\\
 +
\end{pmatrix}\label{x}
 +
$$
 +
consisting of $m$ rows and $n$ columns, the
 +
entries $a_{ij}$ of which belong to some set $K$. (1) is called also an
 +
$(m\times n)$-dimensional matrix over $K$, or a matrix of dimensions $m\times n$ over
 +
$K$. Let $\def\M{\mathrm{M}}\M_{m,n}(K)$ denote the set of all $(m\times n)$-dimensional matrices over
 +
$K$. If $m=n$, then (1) is called a square matrix of order $n$. The set
 +
of all square matrices of order $n$ over $K$ is denoted by $\M_n(K)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
Alternative notations for matrices are:  
 +
$$\begin{bmatrix}
 +
a_{11} & \dots & a_{1n}\\
 +
\dots  & \dots & \dots \\
 +
a_{m1} & \dots & a_{mn}
 +
\end{bmatrix},\quad 
 +
\begin{Vmatrix}
 +
a_{11} & \dots & a_{1n}\\
 +
\dots  & \dots & \dots \\
 +
a_{m1} & \dots & a_{mn}
 +
\end{Vmatrix}, {\rm\ \ and\ }\quad (a_{ij}).
 +
$$
 +
In the most important
 +
cases the role of $K$ is played by the field of real numbers, the
 +
field of complex numbers, an arbitrary field, a ring of polynomials,
 +
the ring of integers, a ring of functions, or an arbitrary associative
 +
ring. The operations of addition and multiplication defined on $K$ are
 +
carried over naturally to matrices over $K$, and in this way one is
 +
led to the matrix calculus — the subject matter of the theory of
 +
matrices.
  
consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627802.png" /> rows and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627803.png" /> columns, the entries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627804.png" /> of which belong to some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627805.png" />. (1) is called also an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627807.png" />-dimensional matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m0627808.png" />, or a matrix of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278010.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278011.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278012.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278013.png" />-dimensional matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278015.png" />, then (1) is called a square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278017.png" />. The set of all square matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278018.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278019.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278020.png" />.
+
The notion of a matrix arose first in the middle of the 19th century
 
+
in the investigations of W. Hamilton, and A. Cayley. Fundamental
Alternative notations for matrices are:
+
results in the theory of matrices are due to K. Weierstrass, C. Jordan
 
+
and G. Frobenius. I.A. Lappo-Danilevskii has developed the theory of
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278021.png" /></td> </tr></table>
+
analytic functions of several matrix variables and has applied it to
 
+
the study of systems of linear differential equations.
In the most important cases the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278022.png" /> is played by the field of real numbers, the field of complex numbers, an arbitrary field, a ring of polynomials, the ring of integers, a ring of functions, or an arbitrary associative ring. The operations of addition and multiplication defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278023.png" /> are carried over naturally to matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278024.png" />, and in this way one is led to the matrix calculus — the subject matter of the theory of matrices.
 
 
 
The notion of a matrix arose first in the middle of the 19th century in the investigations of W. Hamilton, and A. Cayley. Fundamental results in the theory of matrices are due to K. Weierstrass, C. Jordan and G. Frobenius. I.A. Lappo-Danilevskii has developed the theory of analytic functions of several matrix variables and has applied it to the study of systems of linear differential equations.
 
  
 
==Operations with matrices.==
 
==Operations with matrices.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278025.png" /> be an associative ring and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278026.png" />. Then the sum of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278028.png" /> is, by definition,
+
Let $K$ be an associative ring and let
 +
$A=(a_{ij}),\; B=(b_{ij})\in \M_{m,n}(K)$. Then the sum of the matrices $A$ and $B$ is, by definition,
 +
$$A+B = (a_{ij}+b{ij}).$$
 +
Clearly, $A+B\in\M_{m,n}(K)$ and addition of matrices is associative and
 +
commutative. The null matrix in $\M_{m,n}(K)$ is the matrix 0, all entries of
 +
which are zero. For every $A\in\M_{m,n}(K)$,
 +
$$A+0=0+A=A$$
 +
Let $A=(a_{ij})\in\M_{m,k}(K)$ and $B=(b_{ij})\in\M_{k,n}(K)$. The product of
 +
the two matrices $A$ and $B$ is defined by the rule
 +
$$AB=(c_{\mu,\nu})\in\M_{m,n}(K)$$
 +
where
 +
$$c_{\mu,\nu} = \sum_{j=1}^k a_{\mu j}b_{j\nu}.$$
 +
The product of two elements of $\M_n(K)$ is always defined and belongs to
 +
$\M_n(K)$. Multiplication of matrices is associative: If $A\in\M_{m,k}(K)$, $B\in\M_{k,n}(K)$ and $C\in\M_{n,p}(K)$,
 +
then
 +
$$(AB)C = A(BC)$$
 +
and $ABC\in\M_{m,p}(K)$. The distributivity rule also holds: For $A\in\M_{m,n}(K)$ and
 +
$B,C\in\M_{n,m}(K)$,
 +
$$A(B+C)=AB+AC,\quad (B+C)A=BA+CA.\label{y}$$
 +
In particular, (2) holds also for $A,B,C\in\M_{n}(K)$. Consequently, $\M_{n}(K)$ is
 +
an associative ring. If $K$ is a ring with an identity, then the
 +
matrix
 +
$$\def\E{\mathrm{E}}\E_n = \begin{pmatrix}1&\dots&0\\
 +
\vdots&\ddots&\vdots\\ 0&\dots&1\end{pmatrix}$$
 +
is the identity of the ring $\M_{n}(K)$:
 +
$$\E_n A = A\E_n = A$$
 +
for all
 +
$A\in\M_{n}(K)$. Multiplication of matrices is not commutative: If $n\ge 2$, for every
 +
associative ring $K$ with an identity there are matrices $A,B\in\M_{n}(K)$ such that
 +
$AB\ne BA$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278029.png" /></td> </tr></table>
+
Let $\alpha\in K$, $A=(a_{ij})\in\M_{m,n}(K)$; the product of the matrix $A$ by the element (number,
 
+
scalar) $\alpha$ is, by definition, the matrix $\alpha A= (\alpha a_{ij})$. Then
Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278030.png" /> and addition of matrices is associative and commutative. The null matrix in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278031.png" /> is the matrix 0, all entries of which are zero. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278032.png" />,
+
$$(\alpha+\beta)A = \alpha A+\beta B,\quad \alpha(\beta A)=(\alpha\beta)A, \quad\alpha (A+B) = \alpha A+\beta B.$$
 
+
Let $K$ be a
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278033.png" /></td> </tr></table>
+
ring with an identity. The matrix $e_{ij}$ is defined as the element of $\M_{m,n}(K)$
 
+
the only non-zero entry of which is the entry $(i,j)$, which equals 1,
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278035.png" />. The product of the two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278037.png" /> is defined by the rule
+
$1\le i\le m$, $1\le j\le n$. For every $A=(a_{ij})\in \M_{m,n}(K)$,  
 
+
$$A=\sum_{i=1}^m\sum_{j=1}^n a_{ij}e_{ij}.$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278038.png" /></td> </tr></table>
+
If $K$ is a field, then $\M_{m,n}(K)$ is an
 +
$nm$-dimensional
 +
[[Vector space|vector space]] over $K$, and the matrices $e_{ij}$ form a
 +
basis in this space.
  
 +
==Block matrices.==
 +
Let $m=m_1+\dots+m_k$, $n=n_1+\dots+n_l$, where $m_\mu$ and $n_\nu$ are positive
 +
integers. Then a matrix $A\in \M_{m,n}(K)$ can be written in the form
 +
$$A=\begin{pmatrix}A_{11}&\dots&A_{1l}\\
 +
\dots &\dots &\dots \\
 +
A_{k1}&\dots &A_{kl}\\
 +
\end{pmatrix}\label{z}$$
 
where
 
where
 +
$A_{\mu\nu}\in \M_{m_\mu,n_\nu}(K)$, $\mu=1,\dots,k$, $\nu=1,\dots,l$. The matrix (3) is called a block matrix. If $B\in \M_{n,p}(K)$, $p=p_1+\dots+p_t$,
 +
$p_i>0$, and $B$ is written in the form
 +
$$B=\begin{pmatrix}B_{11}&\dots&B_{1t}\\
 +
\dots &\dots &\dots \\
 +
B_{l1}&\dots &B_{lt}\\
 +
\end{pmatrix},\quad B_{ij}\in\M_{n_\nu p_j}(K),$$
 +
then
 +
$$AB=C=\begin{pmatrix}C_{11}&\dots&C_{1t}\\
 +
\dots &\dots &\dots \\
 +
C_{k1}&\dots &B_{kt}\\
 +
\end{pmatrix},\quad C_{\mu j} = \sum_{i=1}^l A_{\mu i}B_{ij}.$$
 +
For example, if
 +
$n=kl$, then $\M_n(K)$ may be regarded as $\M_k(\Sigma)$, where $\Sigma = M_l(K)$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278039.png" /></td> </tr></table>
+
The matrix $A\in\M_n(K)$ of the form
 
+
$$\begin{pmatrix}
The product of two elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278040.png" /> is always defined and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278041.png" />. Multiplication of matrices is associative: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278044.png" />, then
+
A_{1} & 0_{12} & \dots & 0_{1k}\\
 
+
0_{21} & A_{2} & \dots & 0_{2k}\\
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278045.png" /></td> </tr></table>
+
\dots  &\dots  &\dots  & \dots \\
 
+
0_{k1} & 0_{k2} & \dots & A_{k}\\
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278046.png" />. The distributivity rule also holds: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278048.png" />,
+
\end{pmatrix},$$
 
+
where $A_i\in\M_{n_i}(K)$ and $0_{ij}\in \M_{n_i n_j}(K)$ is the null matrix,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
is denoted by $\def\diag{\mathrm{diag}}\diag[A_1,\dots,A_k]$ and is called block diagonal. The following holds:
 
 
In particular, (2) holds also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278050.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278051.png" /> is an associative ring. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278052.png" /> is a ring with an identity, then the matrix
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278053.png" /></td> </tr></table>
 
 
 
is the identity of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278054.png" />:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278055.png" /></td> </tr></table>
+
$$\diag[A_1,\dots,A_k]  +\diag[B_1,\dots,B_k] = \diag[A_1+B_1,\dots,A_k+B_k],$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278056.png" />. Multiplication of matrices is not commutative: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278057.png" />, for every associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278058.png" /> with an identity there are matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278060.png" />.
+
$$\diag[A_1,\dots,A_k]  \diag[B_1,\dots,B_k] = \diag[A_1 B_1,\dots,A_k B_k],$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278062.png" />; the product of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278063.png" /> by the element (number, scalar) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278064.png" /> is, by definition, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278065.png" />. Then
+
provided that the orders of $A_i$ and $B_i$ coincide for $i=1,\dots,k$.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278066.png" /></td> </tr></table>
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278067.png" /> be a ring with an identity. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278068.png" /> is defined as the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278069.png" /> the only non-zero entry of which is the entry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278070.png" />, which equals 1, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278072.png" />. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278073.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278074.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278075.png" /> is a field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278076.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278077.png" />-dimensional [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278078.png" />, and the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278079.png" /> form a basis in this space.
 
 
 
==Block matrices.==
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278083.png" /> are positive integers. Then a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278084.png" /> can be written in the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278088.png" />. The matrix (3) is called a block matrix. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278089.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278091.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278092.png" /> is written in the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278093.png" /></td> </tr></table>
 
 
 
then
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278094.png" /></td> </tr></table>
 
 
 
For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278095.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278096.png" /> may be regarded as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278098.png" />.
 
 
 
The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m06278099.png" /> of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780100.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780102.png" /> is the null matrix, is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780103.png" /> and is called block diagonal. The following holds:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780104.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780105.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780106.png" /></td> </tr></table>
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780107.png" /></td> </tr></table>
 
 
 
provided that the orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780109.png" /> coincide for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780110.png" />.
 
  
 
==Square matrices over a field.==
 
==Square matrices over a field.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780111.png" /> be a field, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780112.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780113.png" /> be the [[Determinant|determinant]] of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780114.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780115.png" /> is said to be non-degenerate (or non-singular) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780116.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780117.png" /> is called the inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780118.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780119.png" />. The invertibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780120.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780121.png" /> is equivalent to its non-degeneracy, and
+
Let $K$ be a field, let $A\in\M_n(K)$ and let
 
+
$\det A$ be the
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780122.png" /></td> </tr></table>
+
[[Determinant|determinant]] of the matrix $A$. $A$ is said to be
 
+
non-degenerate (or non-singular) if $\det A \ne 0$. A matrix $A^{-1}\in\M_n(K)$ is called the
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780123.png" /> is the [[Cofactor|cofactor]] of the entry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780125.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780126.png" />,
+
inverse of $A$ if $AA^{-1}=A^{-1} A = \E_n$. The invertibility of $A$ in $\M_n(K)$ is equivalent
 
+
to its non-degeneracy, and  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780127.png" /></td> </tr></table>
+
$$A^{-1} = (c_{ij}),\quad c_{ij} = \frac{A_{ji}}{\det A},$$
 
+
where $A_{ij}$ is the
The set of all invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780128.png" /> is a group under multiplication, called the [[General linear group|general linear group]] and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780129.png" />. The powers of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780130.png" /> are defined as follows
+
[[Cofactor|cofactor]] of the entry $a_{ji}$, $\det(A^{-1})=(\det A)^{-1}$. For $A,B \in\M_n(K)$,  
 
+
$$AB=\E_n \Leftrightarrow BA=\E_n$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780131.png" /></td> </tr></table>
+
The set of
 
+
all invertible elements of $\M_n(K)$ is a group under multiplication, called
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780132.png" /></td> </tr></table>
+
the
 
+
[[General linear group|general linear group]] and denoted by $\def\GL{\mathrm{GL}}\GL(n,K)$. The
and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780133.png" /> is invertible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780134.png" />. For the polynomial
+
powers of a matrix $A$ are defined as follows  
 
+
$$A^0 = \E_n,$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780135.png" /></td> </tr></table>
 
 
 
the matrix polynomial
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780136.png" /></td> </tr></table>
 
  
 +
$$A^k = A^{k-1} A {\rm\ for\ } k>0,$$
 +
and if $A$ is invertible, then $A^{-k} = (A^{-1})^k$. For the polynomial
 +
$$f(x) = \alpha_0 \E_n + \alpha_1 x + \dots + \alpha_t x^t,\quad f(x)\in K[x],$$
 +
the
 +
matrix polynomial
 +
$$f(A) = \alpha_0 \E_n + \alpha_1 A + \dots + \alpha_t A^t$$
 
is defined.
 
is defined.
  
Every matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780137.png" /> gives rise to a linear transformation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780138.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780139.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780140.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780141.png" /> be a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780142.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780143.png" /> be a linear transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780144.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780145.png" /> is uniquely determined by the set of vectors
+
Every matrix from $\M_n(K)$ gives rise to a linear transformation of the
 +
$n$-dimensional vector space $V$ over $K$. Let $v_1,\dots v_n$ be a basis in $V$
 +
and let $\sigma:V\to V$ be a linear transformation of $V$. Then
 +
$\sigma$ is uniquely
 +
determined by the set of vectors
 +
$$u_1=\sigma(v_1),\dots,u_n=\sigma(v_n).$$
 +
Moreover,
 +
$$\displaylines{\sigma v_1 = v_1 a_{11} +\cdots+v_n a_{n1},\cr
 +
\cdots\cdots\cr
 +
\sigma v_1 = v_1 a_{1n} +\cdots+v_n a_{nn},}$$
 +
where $a_{ij}\in K$. The
 +
matrix $A=(a_{ij})$ is called the matrix of the transformation $\sigma$ in the basis
 +
$v_1,\dots,v_n$. For a fixed basis, the matrix $A+B$ is the matrix of the linear
 +
transformation $\sigma+\tau$, while $AB$ is the matrix of $\sigma\tau$ if $B$ is the
 +
matrix of the linear transformation $\tau$. Equality (4) may be written
 +
in the form
 +
$$[\sigma(v_1),\dots,\sigma(v_n)] = [v_1,\dots,v_n]A. $$
 +
Suppose that $w_1,\dots,w_n$ is a second basis in $V$. Then $[w_1,\dots,w_n]=[v_1,\dots,v_n] T$,
 +
$T\in\GL(n,K)$, and $T^{-1}AT$ is the matrix of the transformation $\sigma$ in the basis
 +
$[w_1,\dots,w_n]$. Two matrices $A,B\in\M_n(K)$ are similar if there is a matrix $T\in\GL(n,K)$ such that
 +
$B = T^{-1}AT$. Here, also, $\det A = \det\; T^{-1}AT$ and the ranks of the matrices $A$ and $B$
 +
coincide. The linear transformation $\sigma$ is called non-degenerate, or
 +
non-singular, if $\sigma(V)=V$; $\sigma$ is non-degenerate if and only if its matrix
 +
is non-degenerate. If $V$ is regarded as the space of columns $\M_{n,1}(K)$,
 +
then every linear transformation in $V$ is given by left
 +
multiplication of the columns $\nu\in V$ by some $A\in\M_n(K)$: $\sigma(v)=Av$, and the matrix of
 +
$\sigma$ in the basis
 +
$$v_1 = \begin{pmatrix}1\\0\\ \vdots\\0\end{pmatrix},\dots,
 +
      v_n = \begin{pmatrix}0\\ \vdots\\0\\1\end{pmatrix}$$
 +
coincides with $A$. A matrix $A\in\M_n(K)$ is singular
 +
(or degenerate) if and only if there is a column $v\in\M_{n,1}(K)$, $v\ne 0$, such that
 +
$Av=0$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780146.png" /></td> </tr></table>
+
==Transposition and matrices of special form.==
 
+
Let $A=(a_{ij})\in \M_{m,n}(K)$. Then the
Moreover,
+
matrix $A^T = (a_{ij})^T\in \M_{n,m}(K)$, where $a_{ij}^T = a_{ji}$, is called the transpose of $A$. Alternative
 
+
notations are ${}^tA$ and $A'$. Let $A=(a_{ij})\in \M_{m,n}(\C)$. Then $\bar A = (\bar a_{ij})$, where $\bar a_{ij}$ is the complex
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780147.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
conjugate of the number $a_{ij}$, is called the complex conjugate of
 
+
$A$. The matrix $A^* = {\bar A}^T$, where $A\in\M_n(\C)$, is called the Hermitian conjugate of
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780148.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780149.png" /> is called the matrix of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780150.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780151.png" />. For a fixed basis, the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780152.png" /> is the matrix of the linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780153.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780154.png" /> is the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780155.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780156.png" /> is the matrix of the linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780157.png" />. Equality (4) may be written in the form
+
$A$. Many matrices used in applications are given special names:''''''
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780158.png" /></td> </tr></table>
 
 
 
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780159.png" /> is a second basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780160.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780162.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780163.png" /> is the matrix of the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780164.png" /> in the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780165.png" />. Two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780166.png" /> are similar if there is a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780167.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780168.png" />. Here, also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780169.png" /> and the ranks of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780170.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780171.png" /> coincide. The linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780172.png" /> is called non-degenerate, or non-singular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780173.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780174.png" /> is non-degenerate if and only if its matrix is non-degenerate. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780175.png" /> is regarded as the space of columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780176.png" />, then every linear transformation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780177.png" /> is given by left multiplication of the columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780178.png" /> by some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780179.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780180.png" />, and the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780181.png" /> in the basis
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780182.png" /></td> </tr></table>
 
 
 
coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780183.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780184.png" /> is singular (or degenerate) if and only if there is a column <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780185.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780186.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780187.png" />.
 
  
==Transposition and matrices of special form.==
+
{| cellspacing="20" align="center"
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780188.png" />. Then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780189.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780190.png" />, is called the transpose of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780191.png" />. Alternative notations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780192.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780193.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780194.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780195.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780196.png" /> is the complex conjugate of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780197.png" />, is called the complex conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780198.png" />. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780199.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780200.png" />, is called the Hermitian conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780201.png" />. Many matrices used in applications are given special names:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">name of the matrix</td> <td colname="2" style="background-color:white;" colspan="1">defining condition</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">symmetric</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780202.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">skew-symmetric</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780203.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">orthogonal</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780204.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">Hermitian</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780205.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">unitary</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780206.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">normal</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780207.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">unipotent</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780208.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">stochastic</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780209.png" />,</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780210.png" />,</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780211.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780212.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1">doubly-stochastic</td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780213.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780214.png" /> are stochastic</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780215.png" />-matrix</td> <td colname="2" style="background-color:white;" colspan="1">every entry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780216.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780217.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780218.png" /></td> </tr> </tbody> </table>
+
|-
 +
| '''Name of the matrix:''' || '''Defining condition:'''
 +
|-
 +
| symmetric     || $A^T = -A$
 +
|-
 +
| skew-symmetric || $A^T = -A$
 +
|-
 +
| orthogonal     ||  $A^T = A^{-1}$
 +
|-
 +
| Hermitian     ||  $A^* = A$
 +
|-
 +
| unitary       ||  $A^* = A^{-1}$
 +
|-
 +
| normal         ||  $A^*A = A A^*$
 +
|-
 +
| unipotent     ||  $(A-\mathrm{E}_n)^n = 0$
 +
|-
 +
| stochastic     || $A = (a_{ij})\in \mathrm{M}_n(\C^n),\; a_{ij}\ge0,\; \sum_{j=1}^n = 1, \ {\rm for }\  i=1,\dots,n$
 +
|-
 +
| doubly-stochastic || $A$ and $A^T$ are stochastic
 +
|-
 +
| $(0,1)$-matrix || every entry of $A$ is either $0$ or $1$
 +
|-
 +
|}
  
</td></tr> </table>
 
  
 
==Polynomial matrices.==
 
==Polynomial matrices.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780219.png" /> be a field and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780220.png" /> be the ring of all polynomials in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780221.png" /> with coefficients from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780222.png" />. A matrix over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780223.png" /> is called a polynomial matrix. For the elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780224.png" /> one introduces the following elementary operations: 1) multiplication of a row or column of a matrix by a non-zero element of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780225.png" />; and 2) addition to a row (column) of another row (respectively, column) of the given matrix, multiplied by a polynomial from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780226.png" />. Two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780227.png" /> are called equivalent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780228.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780229.png" /> can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780230.png" /> through a finite number of elementary operations.
+
Let $K$ be a field and let $K[x]$ be the ring of
 +
all polynomials in the variable $x$ with coefficients from $K$. A
 +
matrix over $K[x]$ is called a polynomial matrix. For the elements of the
 +
ring $M_n(K[x])$ one introduces the following elementary operations: 1)
 +
multiplication of a row or column of a matrix by a non-zero element of
 +
the field $K$; and 2) addition to a row (column) of another row
 +
(respectively, column) of the given matrix, multiplied by a polynomial
 +
from $K[x]$. Two matrices $A,B\in\M_n(K[x])$ are called equivalent $(A\sim B)$ if $B$ can be
 +
obtained from $A$ through a finite number of elementary operations.
  
Let
+
Let  
 +
$$N=\diag[f_1(x),\dots,f_r(x),0,\dots,0]\in M_n(K[x]),$$
 +
where a) $f_i(x)\ne 0$; b) $f_j(x)$ is divisible by $f_i(x)$ for $j>i$; and c) the
 +
coefficient of the leading term in $f_i(x)$ is equal to 1. Then $N$ is
 +
called a canonical polynomial matrix. Every equivalence class of
 +
elements of the ring $\M_n(K[x])$ contains a unique canonical matrix. If $A\sim N$,
 +
where
 +
$$N=\diag[f_1(x),\dots,f_r(x),0,\dots,0]$$
 +
is a canonical matrix, then the polynomials
 +
$$f_1(x),\dots,f_r(x)$$
 +
are
 +
called the invariant factors of $A$; the number $r$ is identical with
 +
the
 +
[[Rank|rank]] of $A$. A matrix $A\in \M_n(K[x])$ has an inverse in $\M_n(K[x])$ if and only
 +
if $A\sim E_n$. The last condition is in turn equivalent to $\det A \in K\setminus 0$. Two matrices
 +
$A,B\in \M_n(K[x])$ are equivalent if and only if
 +
$$B=PAQ,$$
 +
where $P,Q\in \M_n(K[x])$, $P\sim Q\sim E_n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780231.png" /></td> </tr></table>
+
Let $A\in \M_n(K[x])$. The matrix
 +
$$xE_n - A\in \M_n(K[x])$$
 +
is called the characteristic matrix of $A$
 +
and $\det(xE_n-A)$ is called the characteristic polynomial of $A$. For every
 +
polynomial of the form
 +
$$f(x)=\alpha_0+\alpha_1x+\cdots+\alpha_{n-1}x^{n-1}+x^n\in K[x]$$
 +
there is an $F\in \M_n(K[x])$ such that
 +
$$\det(xE_n-A) = f(x)$$
 +
Such is,
 +
for example, the matrix
 +
$$F=\begin{pmatrix}
 +
0 & 0 &\dots& 0 & -\alpha_0\\
 +
1 & 0 &\dots& 0 & -\alpha_1\\
 +
0 & 1 &\dots& 0 & -\alpha_2\\
 +
\ddots&\ddots&\ddots&\vdots&\vdots\\
 +
0 & 0 &\dots& 1 & -\alpha_0\\ \end{pmatrix}$$
 +
The characteristic polynomials of two
 +
similar matrices coincide. However, the fact that two matrices have
 +
identical characteristic polynomials does not necessarily entail the
 +
fact that the matrices are similar. A similarity criterion is: Two
 +
matrices $A,B\in \M_n(K[x])$ are similar if and only if the polynomial matrices $xE_n-A$
 +
and $xE_n-B$ are equivalent. The set of all matrices from $\M_n(K[x])$ having a
 +
given characteristic polynomial $f(x)$ is partitioned into a finite
 +
number of classes of similar matrices; this set reduces to a single
 +
class if and only if $f(x)$ does not have multiple factors in $K[x]$.
  
where a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780232.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780233.png" /> is divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780234.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780235.png" />; and c) the coefficient of the leading term in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780236.png" /> is equal to 1. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780237.png" /> is called a canonical polynomial matrix. Every equivalence class of elements of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780238.png" /> contains a unique canonical matrix. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780239.png" />, where
+
Let $A\in \M_n(K)$, $v\in \M_{n,1}(K)$, $v\ne 0$, and suppose that $Av=\lambda v$, where $\lambda\in K$. Then $v$ is called
 +
an eigen vector of $A$ and $\lambda$ is called an eigen value of $A$. An
 +
element $\lambda\in K$ is an eigen value of a matrix $A$ if and only if it is a
 +
root of the characteristic polynomial of $A$. The set of all columns
 +
$u\in \M_{n,1}(K)$ such that $Au=\lambda u$ for a fixed eigen value $\lambda$ of $A$ is a subspace of
 +
$\M_{n,1}(K)$. The dimension of this subspace equals the defect (or deficiency)
 +
$d$ of the matrix $\lambda \E_n - A$ ($d=n-r$, where $r$ is the rank of $\lambda \E_n - A$). The number
 +
$d$ does not exceed the multiplicity of the root $\lambda$, but need not
 +
coincide with it. A matrix $A\in \M_{n}(K)$ is similar to a diagonal matrix if and
 +
only if it has $n$ linearly independent eigen vectors. If for an $A\in \M_{n}(K)$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780240.png" /></td> </tr></table>
+
$$\det(x\E_n - A) = (x-\lambda_1^{n_1}\cdots(x-\lambda_t^{n_t}),\quad \lambda_j\in K$$
 +
and the roots $\lambda_j$ are distinct, then the following holds: $A$ is
 +
similar to a diagonal matrix if and only if for each $\lambda_j$, $j=1,\dots,t$, the
 +
defect of $\lambda_j \E_n - A$ coincides with $n_j$. In particular, every matrix with $n$
 +
distinct eigen values is similar to a diagonal matrix. Over an
 +
algebraically closed field every matrix from $\M_n(K)$ is similar to some
 +
triangular matrix from $\M_n(K)$. The Hamilton–Cayley theorem: If $f(x)$ is the
 +
characteristic polynomial of a matrix $A$, then $f(A)$ is the null
 +
matrix.
  
is a canonical matrix, then the polynomials
+
By definition, the minimum polynomial of a matrix $A\in \M_{n}(K)$ is the
 +
polynomial $m(x)\in K[x]$ with the properties: $\alpha)$) $m(A)=0$; $\beta$) the coefficient of
 +
the leading term equals 1; and $\gamma$) if $0\ne \psi(x)\in K[x]$ and the degree of $\psi(x)$ is
 +
smaller than the degree of $m(x)$, then $\psi(A)\ne 0$. Every matrix has a unique
 +
minimum polynomial. If $g(x)\in K[x]$ and $g(A)=0$, then the minimum polynomial $m(x)$ of
 +
$A$ divides $g(x)$. The minimum polynomial and the characteristic
 +
polynomial of $A$ coincide with the last invariant factor, and,
 +
respectively, the product of all invariant factors, of the matrix
 +
$x\E_n -A$. The minimum polynomial of $A$ equals
 +
$$\frac{\det(x\E_n -A)}{d_{n-1}(x)}$$
 +
where $d_{n-1}(x)$ is the
 +
[[Greatest common divisor|greatest common divisor]] of the minors (cf.
 +
[[Minor|Minor]]) of order $n-1$ of the matrix $x\E_n-A$. A matrix $A\in \M_n(K)$ is
 +
similar to a diagonal matrix over the field $K$ if and only if its
 +
minimum polynomial is a product of distinct linear factors in the ring
 +
$K[x]$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780241.png" /></td> </tr></table>
+
A matrix $A\in \M_n(K)$ is called nilpotent if $A^k=0$ for some integer $k$. A matrix
 
+
$A$ is nilpotent if and only if $\det(x\E_n-A) = x^n$. Every nilpotent matrix from $M_n(K)$
are called the invariant factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780242.png" />; the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780243.png" /> is identical with the [[Rank|rank]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780244.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780245.png" /> has an inverse in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780246.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780247.png" />. The last condition is in turn equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780248.png" />. Two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780249.png" /> are equivalent if and only if
+
is similar to some triangular matrix with zeros on the diagonal.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780250.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780251.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780252.png" />.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780253.png" />. The matrix
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780254.png" /></td> </tr></table>
 
 
 
is called the characteristic matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780255.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780256.png" /> is called the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780257.png" />. For every polynomial of the form
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780258.png" /></td> </tr></table>
 
 
 
there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780259.png" /> such that
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780260.png" /></td> </tr></table>
 
 
 
Such is, for example, the matrix
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780261.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
 
 
 
The characteristic polynomials of two similar matrices coincide. However, the fact that two matrices have identical characteristic polynomials does not necessarily entail the fact that the matrices are similar. A similarity criterion is: Two matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780262.png" /> are similar if and only if the polynomial matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780263.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780264.png" /> are equivalent. The set of all matrices from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780265.png" /> having a given characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780266.png" /> is partitioned into a finite number of classes of similar matrices; this set reduces to a single class if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780267.png" /> does not have multiple factors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780268.png" />.
 
 
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780269.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780270.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780271.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780272.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780273.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780274.png" /> is called an eigen vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780275.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780276.png" /> is called an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780277.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780278.png" /> is an eigen value of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780279.png" /> if and only if it is a root of the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780280.png" />. The set of all columns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780281.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780282.png" /> for a fixed eigen value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780283.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780284.png" /> is a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780285.png" />. The dimension of this subspace equals the defect (or deficiency) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780286.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780287.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780288.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780289.png" /> is the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780290.png" />). The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780291.png" /> does not exceed the multiplicity of the root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780292.png" />, but need not coincide with it. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780293.png" /> is similar to a diagonal matrix if and only if it has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780294.png" /> linearly independent eigen vectors. If for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780295.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780296.png" /></td> </tr></table>
 
 
 
and the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780297.png" /> are distinct, then the following holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780298.png" /> is similar to a diagonal matrix if and only if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780299.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780300.png" />, the defect of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780301.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780302.png" />. In particular, every matrix with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780303.png" /> distinct eigen values is similar to a diagonal matrix. Over an algebraically closed field every matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780304.png" /> is similar to some triangular matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780305.png" />. The Hamilton–Cayley theorem: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780306.png" /> is the characteristic polynomial of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780307.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780308.png" /> is the null matrix.
 
 
 
By definition, the minimum polynomial of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780309.png" /> is the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780310.png" /> with the properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780311.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780312.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780313.png" />) the coefficient of the leading term equals 1; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780314.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780315.png" /> and the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780316.png" /> is smaller than the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780317.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780318.png" />. Every matrix has a unique minimum polynomial. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780319.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780320.png" />, then the minimum polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780321.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780322.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780323.png" />. The minimum polynomial and the characteristic polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780324.png" /> coincide with the last invariant factor, and, respectively, the product of all invariant factors, of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780325.png" />. The minimum polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780326.png" /> equals
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780327.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780328.png" /> is the [[Greatest common divisor|greatest common divisor]] of the minors (cf. [[Minor|Minor]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780329.png" /> of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780330.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780331.png" /> is similar to a diagonal matrix over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780332.png" /> if and only if its minimum polynomial is a product of distinct linear factors in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780333.png" />.
 
 
 
A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780334.png" /> is called nilpotent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780335.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780336.png" />. A matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780337.png" /> is nilpotent if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780338.png" />. Every nilpotent matrix from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780339.png" /> is similar to some triangular matrix with zeros on the diagonal.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Voevodin,   "Algèbre linéare" , MIR (1976) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.I. Kostrikin,   "Introduction to algebra" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.G. Kurosh,   "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> A.I. Mal'tsev,   "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> I.B. Proskuryakov,   "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> A.R.I. Tyshkevich,   "Linear algebra and analytic geometry" , Minsk (1976) (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> R. Bellman,   "Introduction to matrix analysis" , McGraw-Hill (1970)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> N. Bourbaki,   "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P. Lancaster,   "Theory of matrices" , Acad. Press (1969)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> M. Marcus,   H. Minc,   "A survey of matrix theory and matrix inequalities" , Allyn &amp; Bacon (1964)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD>
 
+
<TD valign="top"> V.V. Voevodin, "Algèbre linéare" , MIR (1976) (Translated from Russian)</TD>
 
+
</TR><TR><TD valign="top">[2]</TD>
 +
<TD valign="top"> F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , '''1''' , Chelsea, reprint (1977) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[3]</TD>
 +
<TD valign="top"> A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[4]</TD>
 +
<TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[5]</TD>
 +
<TD valign="top"> A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[6]</TD>
 +
<TD valign="top"> I.B. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)</TD>
 +
</TR><TR><TD valign="top">[7]</TD>
 +
<TD valign="top"> A.R.I. Tyshkevich, "Linear algebra and analytic geometry" , Minsk (1976) (In Russian)</TD>
 +
</TR><TR><TD valign="top">[8]</TD>
 +
<TD valign="top"> R. Bellman, "Introduction to matrix analysis" , McGraw-Hill (1970)</TD>
 +
</TR><TR><TD valign="top">[9]</TD>
 +
<TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)</TD>
 +
</TR><TR><TD valign="top">[10]</TD>
 +
<TD valign="top"> P. Lancaster, "Theory of matrices" , Acad. Press (1969)</TD>
 +
</TR><TR><TD valign="top">[11]</TD>
 +
<TD valign="top"> M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Allyn &amp; Bacon (1964)</TD>
 +
</TR></table>
  
 
====Comments====
 
====Comments====
The result on canonical polynomial matrices quoted above has a natural generalization to matrices over principal ideal domains. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780340.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780341.png" /> over a principal ideal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780342.png" /> of the form
+
The result on canonical polynomial matrices quoted
 
+
above has a natural generalization to matrices over principal ideal
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780343.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
domains. An $(m\times n)$-matrix $A$ over a principal ideal domain $R$ of the
 
+
form  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780344.png" /> divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780345.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780346.png" />, is said to be in Smith canonical form. Every matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780347.png" /> over a principal ideal domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780348.png" /> is equivalent to one in Smith canonical form in the sense that there are an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780349.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780350.png" /> and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780351.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780352.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780353.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780354.png" /> are invertible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780355.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780356.png" />, respectively, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062780/m062780357.png" /> is in Smith canonical form.
+
$$\diag [d_1,\dots,d_r,0,\dots,0]$$
 +
with $d_i$ divisible by $d_{i+1}$, $i=1,\dots,r-1$, is said to be in Smith
 +
canonical form. Every matrix $A$ over a principal ideal domain $R$ is
 +
equivalent to one in Smith canonical form in the sense that there are
 +
an $(m\times m)$-matrix $P$ and an $(n\times n)$-matrix $Q$ such that $P$ and $Q$ are
 +
invertible in $\M_m(R)$ and $\M_n(R)$, respectively, and such that $PAQ$ is in Smith
 +
canonical form.
  
A matrix of the form (a1) is said to be in companion form, especially in linear systems and control theory where the theory of (polynomial) matrices finds many applications.
+
A matrix of the form (a1) is said to be in companion form, especially
 +
in linear systems and control theory where the theory of (polynomial)
 +
matrices finds many applications.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn,   "Algebra" , '''1''' , Wiley (1974) pp. Sect. 10.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> W.A. Wolovich,   "Linear multivariable systems" , Springer (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R.E. Kalman,   P.L. Falb,   M.A. Arbib,   "Topics in mathematical systems theory" , Prentice-Hall (1969)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD>
 +
  <TD valign="top"> P.M. Cohn, "Algebra" , '''1''' , Wiley (1974) pp. Sect. 10.6</TD>
 +
</TR><TR><TD valign="top">[a2]</TD>
 +
  <TD valign="top"> W.A. Wolovich, "Linear multivariable systems" , Springer (1974)</TD>
 +
</TR><TR><TD valign="top">[a3]</TD>
 +
  <TD valign="top"> R.E. Kalman, P.L. Falb, M.A. Arbib, "Topics in mathematical systems theory" , Prentice-Hall (1969)</TD>
 +
</TR></table>

Revision as of 18:38, 13 November 2011

A rectangular array $$\begin{pmatrix} a_{11}&\dots&a_{1n}\\ \dots&\dots&\dots\\ a_{m1}&\dots&a_{mn}\\ \end{pmatrix}\label{x} $$ consisting of $m$ rows and $n$ columns, the entries $a_{ij}$ of which belong to some set $K$. (1) is called also an $(m\times n)$-dimensional matrix over $K$, or a matrix of dimensions $m\times n$ over $K$. Let $\def\M{\mathrm{M}}\M_{m,n}(K)$ denote the set of all $(m\times n)$-dimensional matrices over $K$. If $m=n$, then (1) is called a square matrix of order $n$. The set of all square matrices of order $n$ over $K$ is denoted by $\M_n(K)$.

Alternative notations for matrices are: $$\begin{bmatrix} a_{11} & \dots & a_{1n}\\ \dots & \dots & \dots \\ a_{m1} & \dots & a_{mn} \end{bmatrix},\quad \begin{Vmatrix} a_{11} & \dots & a_{1n}\\ \dots & \dots & \dots \\ a_{m1} & \dots & a_{mn} \end{Vmatrix}, {\rm\ \ and\ }\quad (a_{ij}). $$ In the most important cases the role of $K$ is played by the field of real numbers, the field of complex numbers, an arbitrary field, a ring of polynomials, the ring of integers, a ring of functions, or an arbitrary associative ring. The operations of addition and multiplication defined on $K$ are carried over naturally to matrices over $K$, and in this way one is led to the matrix calculus — the subject matter of the theory of matrices.

The notion of a matrix arose first in the middle of the 19th century in the investigations of W. Hamilton, and A. Cayley. Fundamental results in the theory of matrices are due to K. Weierstrass, C. Jordan and G. Frobenius. I.A. Lappo-Danilevskii has developed the theory of analytic functions of several matrix variables and has applied it to the study of systems of linear differential equations.

Operations with matrices.

Let $K$ be an associative ring and let $A=(a_{ij}),\; B=(b_{ij})\in \M_{m,n}(K)$. Then the sum of the matrices $A$ and $B$ is, by definition, $$A+B = (a_{ij}+b{ij}).$$ Clearly, $A+B\in\M_{m,n}(K)$ and addition of matrices is associative and commutative. The null matrix in $\M_{m,n}(K)$ is the matrix 0, all entries of which are zero. For every $A\in\M_{m,n}(K)$, $$A+0=0+A=A$$ Let $A=(a_{ij})\in\M_{m,k}(K)$ and $B=(b_{ij})\in\M_{k,n}(K)$. The product of the two matrices $A$ and $B$ is defined by the rule $$AB=(c_{\mu,\nu})\in\M_{m,n}(K)$$ where $$c_{\mu,\nu} = \sum_{j=1}^k a_{\mu j}b_{j\nu}.$$ The product of two elements of $\M_n(K)$ is always defined and belongs to $\M_n(K)$. Multiplication of matrices is associative: If $A\in\M_{m,k}(K)$, $B\in\M_{k,n}(K)$ and $C\in\M_{n,p}(K)$, then $$(AB)C = A(BC)$$ and $ABC\in\M_{m,p}(K)$. The distributivity rule also holds: For $A\in\M_{m,n}(K)$ and $B,C\in\M_{n,m}(K)$, $$A(B+C)=AB+AC,\quad (B+C)A=BA+CA.\label{y}$$ In particular, (2) holds also for $A,B,C\in\M_{n}(K)$. Consequently, $\M_{n}(K)$ is an associative ring. If $K$ is a ring with an identity, then the matrix $$\def\E{\mathrm{E}}\E_n = \begin{pmatrix}1&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&1\end{pmatrix}$$ is the identity of the ring $\M_{n}(K)$: $$\E_n A = A\E_n = A$$ for all $A\in\M_{n}(K)$. Multiplication of matrices is not commutative: If $n\ge 2$, for every associative ring $K$ with an identity there are matrices $A,B\in\M_{n}(K)$ such that $AB\ne BA$.

Let $\alpha\in K$, $A=(a_{ij})\in\M_{m,n}(K)$; the product of the matrix $A$ by the element (number, scalar) $\alpha$ is, by definition, the matrix $\alpha A= (\alpha a_{ij})$. Then $$(\alpha+\beta)A = \alpha A+\beta B,\quad \alpha(\beta A)=(\alpha\beta)A, \quad\alpha (A+B) = \alpha A+\beta B.$$ Let $K$ be a ring with an identity. The matrix $e_{ij}$ is defined as the element of $\M_{m,n}(K)$ the only non-zero entry of which is the entry $(i,j)$, which equals 1, $1\le i\le m$, $1\le j\le n$. For every $A=(a_{ij})\in \M_{m,n}(K)$, $$A=\sum_{i=1}^m\sum_{j=1}^n a_{ij}e_{ij}.$$ If $K$ is a field, then $\M_{m,n}(K)$ is an $nm$-dimensional vector space over $K$, and the matrices $e_{ij}$ form a basis in this space.

Block matrices.

Let $m=m_1+\dots+m_k$, $n=n_1+\dots+n_l$, where $m_\mu$ and $n_\nu$ are positive integers. Then a matrix $A\in \M_{m,n}(K)$ can be written in the form $$A=\begin{pmatrix}A_{11}&\dots&A_{1l}\\ \dots &\dots &\dots \\ A_{k1}&\dots &A_{kl}\\ \end{pmatrix}\label{z}$$ where $A_{\mu\nu}\in \M_{m_\mu,n_\nu}(K)$, $\mu=1,\dots,k$, $\nu=1,\dots,l$. The matrix (3) is called a block matrix. If $B\in \M_{n,p}(K)$, $p=p_1+\dots+p_t$, $p_i>0$, and $B$ is written in the form $$B=\begin{pmatrix}B_{11}&\dots&B_{1t}\\ \dots &\dots &\dots \\ B_{l1}&\dots &B_{lt}\\ \end{pmatrix},\quad B_{ij}\in\M_{n_\nu p_j}(K),$$ then $$AB=C=\begin{pmatrix}C_{11}&\dots&C_{1t}\\ \dots &\dots &\dots \\ C_{k1}&\dots &B_{kt}\\ \end{pmatrix},\quad C_{\mu j} = \sum_{i=1}^l A_{\mu i}B_{ij}.$$ For example, if $n=kl$, then $\M_n(K)$ may be regarded as $\M_k(\Sigma)$, where $\Sigma = M_l(K)$.

The matrix $A\in\M_n(K)$ of the form $$\begin{pmatrix} A_{1} & 0_{12} & \dots & 0_{1k}\\ 0_{21} & A_{2} & \dots & 0_{2k}\\ \dots &\dots &\dots & \dots \\ 0_{k1} & 0_{k2} & \dots & A_{k}\\ \end{pmatrix},$$ where $A_i\in\M_{n_i}(K)$ and $0_{ij}\in \M_{n_i n_j}(K)$ is the null matrix, is denoted by $\def\diag{\mathrm{diag}}\diag[A_1,\dots,A_k]$ and is called block diagonal. The following holds:

$$\diag[A_1,\dots,A_k] +\diag[B_1,\dots,B_k] = \diag[A_1+B_1,\dots,A_k+B_k],$$

$$\diag[A_1,\dots,A_k] \diag[B_1,\dots,B_k] = \diag[A_1 B_1,\dots,A_k B_k],$$

provided that the orders of $A_i$ and $B_i$ coincide for $i=1,\dots,k$.

Square matrices over a field.

Let $K$ be a field, let $A\in\M_n(K)$ and let $\det A$ be the determinant of the matrix $A$. $A$ is said to be non-degenerate (or non-singular) if $\det A \ne 0$. A matrix $A^{-1}\in\M_n(K)$ is called the inverse of $A$ if $AA^{-1}=A^{-1} A = \E_n$. The invertibility of $A$ in $\M_n(K)$ is equivalent to its non-degeneracy, and $$A^{-1} = (c_{ij}),\quad c_{ij} = \frac{A_{ji}}{\det A},$$ where $A_{ij}$ is the cofactor of the entry $a_{ji}$, $\det(A^{-1})=(\det A)^{-1}$. For $A,B \in\M_n(K)$, $$AB=\E_n \Leftrightarrow BA=\E_n$$ The set of all invertible elements of $\M_n(K)$ is a group under multiplication, called the general linear group and denoted by $\def\GL{\mathrm{GL}}\GL(n,K)$. The powers of a matrix $A$ are defined as follows $$A^0 = \E_n,$$

$$A^k = A^{k-1} A {\rm\ for\ } k>0,$$ and if $A$ is invertible, then $A^{-k} = (A^{-1})^k$. For the polynomial $$f(x) = \alpha_0 \E_n + \alpha_1 x + \dots + \alpha_t x^t,\quad f(x)\in K[x],$$ the matrix polynomial $$f(A) = \alpha_0 \E_n + \alpha_1 A + \dots + \alpha_t A^t$$ is defined.

Every matrix from $\M_n(K)$ gives rise to a linear transformation of the $n$-dimensional vector space $V$ over $K$. Let $v_1,\dots v_n$ be a basis in $V$ and let $\sigma:V\to V$ be a linear transformation of $V$. Then $\sigma$ is uniquely determined by the set of vectors $$u_1=\sigma(v_1),\dots,u_n=\sigma(v_n).$$ Moreover, $$\displaylines{\sigma v_1 = v_1 a_{11} +\cdots+v_n a_{n1},\cr \cdots\cdots\cr \sigma v_1 = v_1 a_{1n} +\cdots+v_n a_{nn},}$$ where $a_{ij}\in K$. The matrix $A=(a_{ij})$ is called the matrix of the transformation $\sigma$ in the basis $v_1,\dots,v_n$. For a fixed basis, the matrix $A+B$ is the matrix of the linear transformation $\sigma+\tau$, while $AB$ is the matrix of $\sigma\tau$ if $B$ is the matrix of the linear transformation $\tau$. Equality (4) may be written in the form $$[\sigma(v_1),\dots,\sigma(v_n)] = [v_1,\dots,v_n]A. $$ Suppose that $w_1,\dots,w_n$ is a second basis in $V$. Then $[w_1,\dots,w_n]=[v_1,\dots,v_n] T$, $T\in\GL(n,K)$, and $T^{-1}AT$ is the matrix of the transformation $\sigma$ in the basis $[w_1,\dots,w_n]$. Two matrices $A,B\in\M_n(K)$ are similar if there is a matrix $T\in\GL(n,K)$ such that $B = T^{-1}AT$. Here, also, $\det A = \det\; T^{-1}AT$ and the ranks of the matrices $A$ and $B$ coincide. The linear transformation $\sigma$ is called non-degenerate, or non-singular, if $\sigma(V)=V$; $\sigma$ is non-degenerate if and only if its matrix is non-degenerate. If $V$ is regarded as the space of columns $\M_{n,1}(K)$, then every linear transformation in $V$ is given by left multiplication of the columns $\nu\in V$ by some $A\in\M_n(K)$: $\sigma(v)=Av$, and the matrix of $\sigma$ in the basis $$v_1 = \begin{pmatrix}1\\0\\ \vdots\\0\end{pmatrix},\dots, v_n = \begin{pmatrix}0\\ \vdots\\0\\1\end{pmatrix}$$ coincides with $A$. A matrix $A\in\M_n(K)$ is singular (or degenerate) if and only if there is a column $v\in\M_{n,1}(K)$, $v\ne 0$, such that $Av=0$.

Transposition and matrices of special form.

Let $A=(a_{ij})\in \M_{m,n}(K)$. Then the matrix $A^T = (a_{ij})^T\in \M_{n,m}(K)$, where $a_{ij}^T = a_{ji}$, is called the transpose of $A$. Alternative notations are ${}^tA$ and $A'$. Let $A=(a_{ij})\in \M_{m,n}(\C)$. Then $\bar A = (\bar a_{ij})$, where $\bar a_{ij}$ is the complex conjugate of the number $a_{ij}$, is called the complex conjugate of $A$. The matrix $A^* = {\bar A}^T$, where $A\in\M_n(\C)$, is called the Hermitian conjugate of $A$. Many matrices used in applications are given special names:'

Name of the matrix: Defining condition:
symmetric $A^T = -A$
skew-symmetric $A^T = -A$
orthogonal $A^T = A^{-1}$
Hermitian $A^* = A$
unitary $A^* = A^{-1}$
normal $A^*A = A A^*$
unipotent $(A-\mathrm{E}_n)^n = 0$
stochastic $A = (a_{ij})\in \mathrm{M}_n(\C^n),\; a_{ij}\ge0,\; \sum_{j=1}^n = 1, \ {\rm for }\ i=1,\dots,n$
doubly-stochastic $A$ and $A^T$ are stochastic
$(0,1)$-matrix every entry of $A$ is either $0$ or $1$


Polynomial matrices.

Let $K$ be a field and let $K[x]$ be the ring of all polynomials in the variable $x$ with coefficients from $K$. A matrix over $K[x]$ is called a polynomial matrix. For the elements of the ring $M_n(K[x])$ one introduces the following elementary operations: 1) multiplication of a row or column of a matrix by a non-zero element of the field $K$; and 2) addition to a row (column) of another row (respectively, column) of the given matrix, multiplied by a polynomial from $K[x]$. Two matrices $A,B\in\M_n(K[x])$ are called equivalent $(A\sim B)$ if $B$ can be obtained from $A$ through a finite number of elementary operations.

Let $$N=\diag[f_1(x),\dots,f_r(x),0,\dots,0]\in M_n(K[x]),$$ where a) $f_i(x)\ne 0$; b) $f_j(x)$ is divisible by $f_i(x)$ for $j>i$; and c) the coefficient of the leading term in $f_i(x)$ is equal to 1. Then $N$ is called a canonical polynomial matrix. Every equivalence class of elements of the ring $\M_n(K[x])$ contains a unique canonical matrix. If $A\sim N$, where $$N=\diag[f_1(x),\dots,f_r(x),0,\dots,0]$$ is a canonical matrix, then the polynomials $$f_1(x),\dots,f_r(x)$$ are called the invariant factors of $A$; the number $r$ is identical with the rank of $A$. A matrix $A\in \M_n(K[x])$ has an inverse in $\M_n(K[x])$ if and only if $A\sim E_n$. The last condition is in turn equivalent to $\det A \in K\setminus 0$. Two matrices $A,B\in \M_n(K[x])$ are equivalent if and only if $$B=PAQ,$$ where $P,Q\in \M_n(K[x])$, $P\sim Q\sim E_n$.

Let $A\in \M_n(K[x])$. The matrix $$xE_n - A\in \M_n(K[x])$$ is called the characteristic matrix of $A$ and $\det(xE_n-A)$ is called the characteristic polynomial of $A$. For every polynomial of the form $$f(x)=\alpha_0+\alpha_1x+\cdots+\alpha_{n-1}x^{n-1}+x^n\in K[x]$$ there is an $F\in \M_n(K[x])$ such that $$\det(xE_n-A) = f(x)$$ Such is, for example, the matrix $$F=\begin{pmatrix} 0 & 0 &\dots& 0 & -\alpha_0\\ 1 & 0 &\dots& 0 & -\alpha_1\\ 0 & 1 &\dots& 0 & -\alpha_2\\ \ddots&\ddots&\ddots&\vdots&\vdots\\ 0 & 0 &\dots& 1 & -\alpha_0\\ \end{pmatrix}$$ The characteristic polynomials of two similar matrices coincide. However, the fact that two matrices have identical characteristic polynomials does not necessarily entail the fact that the matrices are similar. A similarity criterion is: Two matrices $A,B\in \M_n(K[x])$ are similar if and only if the polynomial matrices $xE_n-A$ and $xE_n-B$ are equivalent. The set of all matrices from $\M_n(K[x])$ having a given characteristic polynomial $f(x)$ is partitioned into a finite number of classes of similar matrices; this set reduces to a single class if and only if $f(x)$ does not have multiple factors in $K[x]$.

Let $A\in \M_n(K)$, $v\in \M_{n,1}(K)$, $v\ne 0$, and suppose that $Av=\lambda v$, where $\lambda\in K$. Then $v$ is called an eigen vector of $A$ and $\lambda$ is called an eigen value of $A$. An element $\lambda\in K$ is an eigen value of a matrix $A$ if and only if it is a root of the characteristic polynomial of $A$. The set of all columns $u\in \M_{n,1}(K)$ such that $Au=\lambda u$ for a fixed eigen value $\lambda$ of $A$ is a subspace of $\M_{n,1}(K)$. The dimension of this subspace equals the defect (or deficiency) $d$ of the matrix $\lambda \E_n - A$ ($d=n-r$, where $r$ is the rank of $\lambda \E_n - A$). The number $d$ does not exceed the multiplicity of the root $\lambda$, but need not coincide with it. A matrix $A\in \M_{n}(K)$ is similar to a diagonal matrix if and only if it has $n$ linearly independent eigen vectors. If for an $A\in \M_{n}(K)$,

$$\det(x\E_n - A) = (x-\lambda_1^{n_1}\cdots(x-\lambda_t^{n_t}),\quad \lambda_j\in K$$ and the roots $\lambda_j$ are distinct, then the following holds: $A$ is similar to a diagonal matrix if and only if for each $\lambda_j$, $j=1,\dots,t$, the defect of $\lambda_j \E_n - A$ coincides with $n_j$. In particular, every matrix with $n$ distinct eigen values is similar to a diagonal matrix. Over an algebraically closed field every matrix from $\M_n(K)$ is similar to some triangular matrix from $\M_n(K)$. The Hamilton–Cayley theorem: If $f(x)$ is the characteristic polynomial of a matrix $A$, then $f(A)$ is the null matrix.

By definition, the minimum polynomial of a matrix $A\in \M_{n}(K)$ is the polynomial $m(x)\in K[x]$ with the properties: $\alpha)$) $m(A)=0$; $\beta$) the coefficient of the leading term equals 1; and $\gamma$) if $0\ne \psi(x)\in K[x]$ and the degree of $\psi(x)$ is smaller than the degree of $m(x)$, then $\psi(A)\ne 0$. Every matrix has a unique minimum polynomial. If $g(x)\in K[x]$ and $g(A)=0$, then the minimum polynomial $m(x)$ of $A$ divides $g(x)$. The minimum polynomial and the characteristic polynomial of $A$ coincide with the last invariant factor, and, respectively, the product of all invariant factors, of the matrix $x\E_n -A$. The minimum polynomial of $A$ equals $$\frac{\det(x\E_n -A)}{d_{n-1}(x)}$$ where $d_{n-1}(x)$ is the greatest common divisor of the minors (cf. Minor) of order $n-1$ of the matrix $x\E_n-A$. A matrix $A\in \M_n(K)$ is similar to a diagonal matrix over the field $K$ if and only if its minimum polynomial is a product of distinct linear factors in the ring $K[x]$.

A matrix $A\in \M_n(K)$ is called nilpotent if $A^k=0$ for some integer $k$. A matrix $A$ is nilpotent if and only if $\det(x\E_n-A) = x^n$. Every nilpotent matrix from $M_n(K)$ is similar to some triangular matrix with zeros on the diagonal.

References

[1] V.V. Voevodin, "Algèbre linéare" , MIR (1976) (Translated from Russian)
[2] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)
[3] A.I. Kostrikin, "Introduction to algebra" , Springer (1982) (Translated from Russian)
[4] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)
[5] A.I. Mal'tsev, "Foundations of linear algebra" , Freeman (1963) (Translated from Russian)
[6] I.B. Proskuryakov, "Higher algebra. Linear algebra, polynomials, general algebra" , Pergamon (1965) (Translated from Russian)
[7] A.R.I. Tyshkevich, "Linear algebra and analytic geometry" , Minsk (1976) (In Russian)
[8] R. Bellman, "Introduction to matrix analysis" , McGraw-Hill (1970)
[9] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[10] P. Lancaster, "Theory of matrices" , Acad. Press (1969)
[11] M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964)

Comments

The result on canonical polynomial matrices quoted above has a natural generalization to matrices over principal ideal domains. An $(m\times n)$-matrix $A$ over a principal ideal domain $R$ of the form $$\diag [d_1,\dots,d_r,0,\dots,0]$$ with $d_i$ divisible by $d_{i+1}$, $i=1,\dots,r-1$, is said to be in Smith canonical form. Every matrix $A$ over a principal ideal domain $R$ is equivalent to one in Smith canonical form in the sense that there are an $(m\times m)$-matrix $P$ and an $(n\times n)$-matrix $Q$ such that $P$ and $Q$ are invertible in $\M_m(R)$ and $\M_n(R)$, respectively, and such that $PAQ$ is in Smith canonical form.

A matrix of the form (a1) is said to be in companion form, especially in linear systems and control theory where the theory of (polynomial) matrices finds many applications.

References

[a1] P.M. Cohn, "Algebra" , 1 , Wiley (1974) pp. Sect. 10.6
[a2] W.A. Wolovich, "Linear multivariable systems" , Springer (1974)
[a3] R.E. Kalman, P.L. Falb, M.A. Arbib, "Topics in mathematical systems theory" , Prentice-Hall (1969)
How to Cite This Entry:
Matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix&oldid=19628
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article