SINGULAR-VALUES OF COMPANION MATRICES AND BOUNDS ON ZEROS OF POLYNOMIALS

Let C(p) denote the companion matrix of a monic polynomial p with complex coefficients. Then the zeros of p are exactly the eigenvalues of C(p). In this paper, the singular values of C(p) are computed. Applying some basic eigenvalue-singular value majorization relations, several sharp estimates are obtained for the zeros of p in terms of its coefficients. These estimates improve some classical bounds on zeros of polynomials.