Companion matrix

In linear algebra, the companion matrix of the monic polynomial

$p(t)=c_0 + c_1 t + \dots + c_{n-1}t^{n-1} + t^n$

is the square matrix defined as

$C(p)=\begin{bmatrix} 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{n-1} \\ \end{bmatrix}.$

With this convention, and writing the basis as $v_1,\dots,v_n$, one has Cv_i = C^{i − 1}v₁ = v_{i + 1} (for i < n), and v₁ generates V as a K[C]-module: 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 relations.

Characterization

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 A is an n-by-n matrix with entries from some field K, then the following statements are equivalent:

• A is similar to the companion matrix over K of its characteristic polynomial
• the characteristic polynomial of A coincides with the minimal polynomial of A, equivalently the minimal polynomial has degree n
• there exists a cyclic vector v in V = Kⁿ for A, meaning that {v, Av, A²v,...,A^n-1v} is a basis of V. Equivalently, such that V is cyclic as a K[A]-module (and V = K[A] / (p(A))); one says that 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 A. This is the rational canonical form of A.

Diagonalizability

If p(t) has distinct roots λ₁,...,λ_n (the eigenvalues of C(p)), then C(p) is diagonalizable as follows:

$V C(p) V^{-1} = \mbox{diag}(\lambda_1,\dots,\lambda_n)$

where V is the Vandermonde matrix corresponding to the λ's.

Linear recursive sequences

Given a linear recursive sequence with characteristic polynomial

$p(t)=c_0 + c_1 t + \dots + c_{n-1}t^{n-1} + t^n$

the (transpose) companion matrix

$C^T(p)=\begin{bmatrix} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\\ -c_0 & -c_1 & -c_2 & \cdots & -c_{n-1}\\ \end{bmatrix}$

generates the sequence, in the sense that

$C^T\begin{bmatrix}a_k\\ a_{k+1}\\ \vdots \\ a_{k+n-1} \end{bmatrix} = \begin{bmatrix}a_{k+1}\\ a_{k+2}\\ \vdots \\ a_{k+n} \end{bmatrix}.$

It increments the series by 1.