Companion matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial
is the square matrix defined as
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 are equal to .
In this sense, the matrix is the "companion" of the polynomial .
If is an "n"-by-"n" matrix with entries from some field , 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 . This is the rational canonical form of .
Diagonalizability.
If has distinct roots (the eigenvalues of "C"("p")), then "C"("p") is diagonalizable as follows: 
where is the Vandermonde matrix corresponding to the 's.
In that case, traces of powers "m" of readily yield sums of the same powers "m" of all roots of "p"("t"),
If has a non-simple root, then "C"("p") isn't diagonalizable (its Jordan canonical form contains one block for each distinct root).
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 is an eigenvector of this matrix for eigenvalue , when is a root of the characteristic polynomial .
For , and all other , i.e., , this matrix reduces to Sylvester's cyclic , or circulant matrix.