Vieta-Type Formulas for Matrix Polynomials

Abstract

The classical Vieta formulas relate the coefficients of a complex scalar polynomial to the elementary symmetric polynomials of its roots. In this paper, we establish analogous spectral identities for complex matrix polynomials. For a monic matrix polynomial, we prove that the sum of all its eigenvalues equals the sum of the roots of the product of its diagonal scalar polynomials and is also equal to a constant multiple of the sum of the roots of the scalar polynomial obtained by summing its diagonal entries. We further show that the product of the eigenvalues of a monic matrix polynomial is determined by the determinant of its constant coefficient matrix. As a consequence, we recover the matrix Vieta formulas of Fuchs and Schwarz \cite{Fuchs-Schwarz}, for independent solutions of matrix algebraic equations via a density argument. We also derive corresponding identities for non-monic matrix polynomials with nonsingular leading coefficients and show, by means of an example, that the identities established for monic matrix polynomials do not extend directly to the non-monic case.