Companion matrix
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In linear algebra, the companion matrix of the monic polynomial
- <math>
p(t)=c_0 + c_1 t + \dots + c_{n-1}t^{n-1} + t^n <math>
is the square matrix defined as
- <math>C(p)=\begin{bmatrix}
0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \\ \end{bmatrix}.<math>
(While some authors use the transpose of this matrix, Wikipedia uses the above convention.)
The characteristic polynomial as well as the minimal polynomial of C(p) are equal to p; in this sense, the matrix C(p) is the "companion" of the polynomial p.
If the polynomial p(t) has n different zeros λ1,...,λn (the eigenvalues of C(p)), then C(p) is diagonalizable as follows:
- <math>V C(p) V^{-1} = \mbox{diag}(\lambda_1,\dots,\lambda_n)<math>
where V is the Vandermonde matrix corresponding to the λ's.
If A is an n-by-n matrix with entries from some field K, then the following statements are equivalent:
- A is similar to a companion matrix over K
- the characteristic polynomial of A coincides with the minimal polynomial of A
- there exists a vector v in Kn such that {v, Av, A2v,...,An-1v} is a basis of Kn
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 chose so that their polynomials divide each other; then they are uniquely determined by A. This is the rational canonical form of A.