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In [[linear algebra]], the  [[Ferdinand Georg Frobenius|Frobenius]]  '''companion matrix''' of the [[monic polynomial]]
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:<math>
p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n ~,
</math>
is the [[square matrix]] defined as


:<math>C(p)=\begin{bmatrix}
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0 & 0 & \dots & 0 & -c_0 \\
1 & 0 & \dots & 0 & -c_1 \\
0 & 1 & \dots & 0 & -c_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 1 & -c_{n-1}
\end{bmatrix}.</math>


With this convention, and on the [[Basis (linear algebra)|basis]]  {{math| ''v''<sub>1</sub>,  ...  , ''v''<sub>''n''</sub>}}, one has
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:<math>Cv_i = C^{i}v_1 = v_{i+1}</math>
(for {{math|''i'' < ''n''}}), and  {{math| ''v''<sub>1</sub>}}  generates {{mvar|V}} as a {{math| ''K''[''C'']}}-module: {{mvar|C}} cycles basis vectors.


Some authors use the [[transpose]] of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear [[recursive relation]]s.
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==Characterization==
The [[characteristic polynomial]] as well as the [[minimal polynomial (linear algebra)|minimal polynomial]] of {{math|''C''(''p'')}} are equal to {{mvar|p}}.<ref>
{{Cite book | last = Horn | first = Roger A. | coauthors = Charles R. Johnson
| title = Matrix Analysis | publisher = Cambridge University Press | year = 1985
| location = Cambridge, UK | pages = 146–147
| url = http://books.google.com/books?id=f6_r93Of544C&pg=PA147&dq=%22companion+matrix%22&cd=1#v=onepage&q=%22companion%20matrix%22&f=false
| accessdate = 2010-02-10 | isbn = 0-521-30586-1}}</ref>
 
In this sense, the matrix {{math|''C''(''p'')}}  is the "companion" of the polynomial  {{mvar|p}}.
 
If {{mvar|A}}  is an ''n''-by-''n'' matrix with entries from some [[field (mathematics)|field]] {{mvar|K}}, then the following statements are equivalent:
* {{mvar|A}}  is [[similar (linear algebra)|similar]] to the companion matrix over {{mvar|K}}  of its characteristic polynomial
* the characteristic polynomial of {{mvar|A}}  coincides with the minimal polynomial of {{mvar|A}}, equivalently the minimal polynomial has degree {{mvar|n}}
* there exists a [[cyclic permutation|cyclic vector]] {{math|'''v'''}} in <math>V=K^n</math> for {{mvar|A}}, meaning that {'''v''', ''A'''''v''', ''A''<sup>2</sup>'''v''', ..., ''A''<sup>''n''&minus;1</sup>'''v'''} is a [[basis (linear algebra)|basis]] of ''V''. Equivalently, such that ''V'' is cyclic as a <math>K[A]</math>-module (and <math>V=K[A]/(p(A))</math>); one says that {{mvar|A}}  is ''regular''.
 
Not every square matrix is similar to a companion matrix. But every matrix is similar to a matrix made up of blocks of companion matrices. Furthermore, these companion matrices can be chosen so that their polynomials divide each other; then they are uniquely determined by {{mvar|A}}. This is the [[Frobenius normal form|rational canonical form]] of {{mvar|A}}.
 
==Diagonalizability==
If {{math|''p''(''t'')}} has distinct roots {{math|''λ''<sub>1</sub>,&nbsp;...,&nbsp;''λ''<sub>''n''</sub>}} (the [[eigenvalue]]s of ''C''(''p'')), then ''C''(''p'') is [[diagonalizable]] as follows:  
:<math>V C(p) V^{-1} = \operatorname{diag}(\lambda_1,\dots,\lambda_n)</math>
where {{mvar|V}} is the [[Vandermonde matrix]] corresponding to the {{mvar|λ}}'s.
 
In that case,<ref>[[Richard E. Bellman|Bellman]], Richard (1987), ''Introduction to Matrix Analysis'', SIAM, ISBN 0898713994 .</ref> traces of powers ''m'' of {{mvar|C}}  readily yield sums of the same powers ''m'' of all roots of  ''p''(''t''),
:<math>\mathrm{Tr}  C^m = \sum_{i=1}^{n}  \lambda_i^m ~. </math>
 
In general, the companion matrix may be non-diagonalizable.
 
==Linear recursive sequences==
Given a [[linear recursive sequence]] with characteristic polynomial
:<math>p(t)=c_0 + c_1 t + \cdots + c_{n-1}t^{n-1} + t^n \, </math>
the (transpose) companion matrix
:<math>C^T(p)=\begin{bmatrix}
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1\\
-c_0 & -c_1 & -c_2 & \cdots & -c_{n-1}
\end{bmatrix}</math>
generates the sequence, in the sense that
:<math>C^T\begin{bmatrix}a_k\\
a_{k+1}\\
\vdots \\
a_{k+n-1}
\end{bmatrix}
= \begin{bmatrix}a_{k+1}\\
a_{k+2}\\
\vdots \\
a_{k+n}
\end{bmatrix}.</math>
increments the series by 1.
 
The vector {{math|(1,''t'',''t''<sup>2</sup>, ..., ''t''<sup>''n''-1</sup>)}} is an eigenvector of this matrix for eigenvalue {{mvar|t}}, when  {{mvar|t}} is a root of the characteristic polynomial  {{math|''p''(''t'')}}.
 
For {{math|''c''<sub>0</sub> {{=}} −1}}, and all other {{math|''c<sub>i</sub>''{{=}}0}},  i.e., {{math|''p''(''t'') {{=}} ''t<sup>n</sup>''−1}},  this matrix reduces to Sylvester's cyclic [[Generalizations_of_Pauli_matrices#Construction:_The_clock_and_shift_matrices|shift matrix]], or [[circulant matrix]].
 
==See also==
*[[Frobenius endomorphism]]
*[[Cayley-Hamilton theorem]]
 
==Notes==
{{reflist}}
 
{{DEFAULTSORT:Companion Matrix}}
[[Category:Matrices]]
[[Category:Matrix theory]]

Latest revision as of 09:04, 29 December 2014

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