Companion matrix
From Wikipedia, the free encyclopedia
In linear algebra, the companion matrix of the monic polynomial
is the square matrix defined as
(While some authors use the transpose of this matrix, Wikipedia uses the above convention.)
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 the polynomial p(t) has n different zeros 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.
If A is an n-by-n matrix with entries from some field K, then the following statements are equivalent:
- A is similar to a companion matrix over K
- the characteristic polynomial of A coincides with the minimal polynomial of A
- there exists a vector v in Kn such that {v, Av, A2v,...,An-1v} is a basis of Kn
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.
