In linear algebra, the Frobenius companion matrix of the monic polynomial
is the square matrix defined as
With this convention, and on the basis v1, ... , vn, one has
(for i < n), and v1 generates Template:Mvar as a K[C]-module: Template:Mvar cycles basis vectors.
Some authors use the transpose of this matrix, which (dually) cycles coordinates, and is more convenient for some purposes, like linear recurrence relations.
The characteristic polynomial as well as the minimal polynomial of C(p) are equal to Template:Mvar.
In this sense, the matrix C(p) is the "companion" of the polynomial Template:Mvar.
If Template:Mvar is an n-by-n matrix with entries from some field Template:Mvar, then the following statements are equivalent:
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 Template:Mvar. This is the rational canonical form of Template:Mvar.
If p(t) has distinct roots λ1, ..., λn (the eigenvalues of C(p)), then C(p) is diagonalizable as follows:
where Template:Mvar is the Vandermonde matrix corresponding to the Template:Mvar's.
In that case, traces of powers m of Template:Mvar readily yield sums of the same powers m of all roots of p(t),
In general, the companion matrix may be non-diagonalizable.
Linear recursive sequences
Given a linear recursive sequence with characteristic polynomial
the (transpose) companion matrix
generates the sequence, in the sense that
increments the series by 1.
The vector (1,t,t2, ..., tn-1) is an eigenvector of this matrix for eigenvalue Template:Mvar, when Template:Mvar is a root of the characteristic polynomial p(t).
For c0 = −1, and all other ci=0, i.e., p(t) = tn−1, this matrix reduces to Sylvester's cyclic shift matrix, or circulant matrix.
- ↑ Bellman, Richard (1987), Introduction to Matrix Analysis, SIAM, ISBN 0898713994 .