A GENERALIZATION OF THE INERTIA THEOREM FOR QUADRATIC MATRIX POLYNOMIALS

We show that the inertia of a quadratic matrix polynomial is determine d in terms of the inertia of its coefficient matrices if the leading c oefficient is Hermitian and nonsingular, the constant term is Hermitia n, and the real part of the coefficient matrix of the first degree ter m is definite. In particular, we prove that the number of zero eigenva lues of such a matrix polynomial is the same as the number of zero eig envalues of its constant term. We also give some new results for the c ase where the real part of the coefficient matrix of the first degree term is semidefinite. (C) 1998 Elsevier Science Inc. All rights reserv ed.