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, ... , v_n, 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.

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:

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.