Companion matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial 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-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. 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.