Companion Matrix - Characterization

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 for A, meaning that {v, Av, A2v, ..., An−1v} is a basis of V. Equivalently, such that V is cyclic as a -module (and ); 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.