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''companion matrix''
 
''companion matrix''
  
For every [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202001.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202002.png" />-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202003.png" /> such that the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202005.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202006.png" />. Indeed, two such are:
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For every [[Polynomial|polynomial]] $f = \lambda ^ { n } + a _ { n - 1 } \lambda ^ { n - 1 } + \ldots + a _ { 1 } \lambda + a _ { 0 }$ there are $( n \times n )$-matrices $A$ such that the [[Characteristic polynomial|characteristic polynomial]] of $A$, $\operatorname { det } ( \lambda I - A )$, is equal to $f$. Indeed, two such are:
  
<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/f/f120/f120200/f1202007.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \left( \begin{array} { c c c c } { 0 } &amp; { \square } &amp; { \square } &amp; { - a _ { 0 } } \\ { 1 } &amp; { \ddots } &amp; { \square } &amp; { - a _ { 1 } } \\ { \square } &amp; { \ddots } &amp; { 0 } &amp; { \vdots } \\ { \square } &amp; { \square } &amp; { 1 } &amp; { - a _ { n - 1 } } \end{array} \right) \end{equation}
  
 
and
 
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/f/f120/f120200/f1202008.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} \left( \begin{array} { c c c c } { 0 } &amp; { 1 } &amp; { \square } &amp; { \square } \\ { \square } &amp; { \ddots } &amp; { \ddots } &amp; { \square } \\ { \square } &amp; { \square } &amp; { 0 } &amp; { 1 } \\ { - a _ { 0 } } &amp; { \cdots } &amp; { \cdots } &amp; { - a _ { n - 1 } } \end{array} \right). \end{equation}
  
These two matrices are similar and their minimal polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f1202009.png" />, i.e. their similarity invariants are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020010.png" /> (see [[Normal form|Normal form]]). Both are called the companion matrix, or Frobenius matrix, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020011.png" />.
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These two matrices are similar and their minimal polynomial (cf. [[Minimal polynomial of a matrix|Minimal polynomial of a matrix]]) is $f$, i.e. their similarity invariants are $1 , \dots , f$ (see [[Normal form|Normal form]]). Both are called the companion matrix, or Frobenius matrix, of $f$.
  
 
More generally, a matrix of block-triangular form with as diagonal blocks one of the companion matrices above (all of the same type),
 
More generally, a matrix of block-triangular form with as diagonal blocks one of the companion matrices above (all of the same type),
  
<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/f/f120/f120200/f12020012.png" /></td> </tr></table>
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\begin{equation*} \left( \begin{array} { c c c } { A _ { 1 } } &amp; { \square } &amp; { * } \\ { \square } &amp; { \ddots } &amp; { \square } \\ { 0 } &amp; { \square } &amp; { A _ { n } } \end{array} \right) \end{equation*}
  
 
is also sometimes called a Frobenius matrix.
 
is also sometimes called a Frobenius matrix.
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Somewhat related, a matrix with just one column (or one row, but not both) different from the identity matrix is also sometimes called a Frobenius matrix; see, e.g., [[#References|[a1]]], p. 169.
 
Somewhat related, a matrix with just one column (or one row, but not both) different from the identity matrix is also sometimes called a Frobenius matrix; see, e.g., [[#References|[a1]]], p. 169.
  
For the matrix (a1), the first standard basis vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020013.png" /> is a [[Cyclic vector|cyclic vector]] (see also [[Pole assignment problem|Pole assignment problem]]). The vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020014.png" /> form a so-called Krylov sequence of vectors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020015.png" />, that is, a sequence of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020018.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020019.png" /> are independent, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020020.png" /> is a linear combination of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020021.png" />.
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For the matrix (a1), the first standard basis vector $e_1$ is a [[Cyclic vector|cyclic vector]] (see also [[Pole assignment problem|Pole assignment problem]]). The vectors $e _ { 1 } , \dots , e _ { n } , - ( a _ { 0 } e _ { 1 } + \ldots + a _ { n  - 1} e _ { n } )$ form a so-called Krylov sequence of vectors for $A$, that is, a sequence of vectors $v _ { 1 } , \dots , v _ { n + 1 }$ such that $A v _ { i } = v _ { i  + 1}$, $i = 1 , \dots , n$, the $v _ { 1 } , \dots , v _ { n }$ are independent, and $v _ { n+1 } = A v _ { n}$ is a linear combination of $v _ { 1 } , \dots , v _ { n }$.
  
The first and second natural canonical forms of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020022.png" /> (see also [[Normal form|Normal form]]) are block-diagonal with companion matrices as blocks. Both are also known as the Frobenius normal form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020023.png" />.
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The first and second natural canonical forms of a matrix $A$ (see also [[Normal form|Normal form]]) are block-diagonal with companion matrices as blocks. Both are also known as the Frobenius normal form of $A$.
  
In a completely different setting, the phrase  "Frobenius matrix"  refers to a matrix giving the (induced) action of the [[Frobenius endomorphism|Frobenius endomorphism]] of an algebraic variety of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120200/f12020024.png" /> on, say, the cohomology of that variety.
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In a completely different setting, the phrase  "Frobenius matrix"  refers to a matrix giving the (induced) action of the [[Frobenius endomorphism|Frobenius endomorphism]] of an algebraic variety of characteristic $p &gt; 0$ on, say, the cohomology of that variety.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Stoer,  R. Bulirsch,  "Introduction to linear algebra" , Springer  (1993)  pp. Sect. 6.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Marcus,  H. Minc,  "A survey of matrix theory and matrix inequalities" , Dover  (1992)  pp. Sect. I.3</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  J. Stoer,  R. Bulirsch,  "Introduction to linear algebra" , Springer  (1993)  pp. Sect. 6.3</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Marcus,  H. Minc,  "A survey of matrix theory and matrix inequalities" , Dover  (1992)  pp. Sect. I.3</td></tr></table>

Latest revision as of 15:30, 1 July 2020

companion matrix

For every polynomial $f = \lambda ^ { n } + a _ { n - 1 } \lambda ^ { n - 1 } + \ldots + a _ { 1 } \lambda + a _ { 0 }$ there are $( n \times n )$-matrices $A$ such that the characteristic polynomial of $A$, $\operatorname { det } ( \lambda I - A )$, is equal to $f$. Indeed, two such are:

\begin{equation} \tag{a1} \left( \begin{array} { c c c c } { 0 } & { \square } & { \square } & { - a _ { 0 } } \\ { 1 } & { \ddots } & { \square } & { - a _ { 1 } } \\ { \square } & { \ddots } & { 0 } & { \vdots } \\ { \square } & { \square } & { 1 } & { - a _ { n - 1 } } \end{array} \right) \end{equation}

and

\begin{equation} \tag{a2} \left( \begin{array} { c c c c } { 0 } & { 1 } & { \square } & { \square } \\ { \square } & { \ddots } & { \ddots } & { \square } \\ { \square } & { \square } & { 0 } & { 1 } \\ { - a _ { 0 } } & { \cdots } & { \cdots } & { - a _ { n - 1 } } \end{array} \right). \end{equation}

These two matrices are similar and their minimal polynomial (cf. Minimal polynomial of a matrix) is $f$, i.e. their similarity invariants are $1 , \dots , f$ (see Normal form). Both are called the companion matrix, or Frobenius matrix, of $f$.

More generally, a matrix of block-triangular form with as diagonal blocks one of the companion matrices above (all of the same type),

\begin{equation*} \left( \begin{array} { c c c } { A _ { 1 } } & { \square } & { * } \\ { \square } & { \ddots } & { \square } \\ { 0 } & { \square } & { A _ { n } } \end{array} \right) \end{equation*}

is also sometimes called a Frobenius matrix.

Somewhat related, a matrix with just one column (or one row, but not both) different from the identity matrix is also sometimes called a Frobenius matrix; see, e.g., [a1], p. 169.

For the matrix (a1), the first standard basis vector $e_1$ is a cyclic vector (see also Pole assignment problem). The vectors $e _ { 1 } , \dots , e _ { n } , - ( a _ { 0 } e _ { 1 } + \ldots + a _ { n - 1} e _ { n } )$ form a so-called Krylov sequence of vectors for $A$, that is, a sequence of vectors $v _ { 1 } , \dots , v _ { n + 1 }$ such that $A v _ { i } = v _ { i + 1}$, $i = 1 , \dots , n$, the $v _ { 1 } , \dots , v _ { n }$ are independent, and $v _ { n+1 } = A v _ { n}$ is a linear combination of $v _ { 1 } , \dots , v _ { n }$.

The first and second natural canonical forms of a matrix $A$ (see also Normal form) are block-diagonal with companion matrices as blocks. Both are also known as the Frobenius normal form of $A$.

In a completely different setting, the phrase "Frobenius matrix" refers to a matrix giving the (induced) action of the Frobenius endomorphism of an algebraic variety of characteristic $p > 0$ on, say, the cohomology of that variety.

References

[a1] J. Stoer, R. Bulirsch, "Introduction to linear algebra" , Springer (1993) pp. Sect. 6.3
[a2] M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Dover (1992) pp. Sect. I.3
How to Cite This Entry:
Frobenius matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_matrix&oldid=49892
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article