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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|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|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|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|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:''''''

{| cellspacing="20" align="center"
|-
| '''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|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|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
$K[x]$.

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====
<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====
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====
<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>