The matrix has the same eigenvalues and right eigenvectors as .
Analogous statements are true about .
Let be a monic polynomial. Define the by companion matrix of as where all entries not explicitly shown are 0. Then the eigenvalues of are the roots of , and the right eigenvectors are . A reliable, but not optimally efficient, algorithm for finding roots of a polynomial is to find all the eigenvalues of . Let be a monic matrix polynomial, where each is an by matrix.