Companion matrix

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 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 recurrence relations.


The characteristic polynomial as well as the minimal polynomial of C(p) are equal to p.[1]

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:

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.


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

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

In that case,[2] traces of powers m of C 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 t, when t 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.

See also


  1. Horn, Roger A.; Charles R. Johnson (1985). Matrix Analysis. Cambridge, UK: Cambridge University Press. pp. 146–147. ISBN 0-521-30586-1. Retrieved 2010-02-10.
  2. Bellman, Richard (1987), Introduction to Matrix Analysis, SIAM, ISBN 0898713994 .
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