B. Bilir et C. Chicone, A GENERALIZATION OF THE INERTIA THEOREM FOR QUADRATIC MATRIX POLYNOMIALS, Linear algebra and its applications, 280(2-3), 1998, pp. 229-240
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.