The inverse of a matrix polynomial

AbstractAn explicit representation is obtained for P(z)−1 when P(z) is a complex n×n matrix polynomial in z whose coefficient of the highest power of z is the identity matrix. The representation is a sum of terms involving negative powers of z−λ for each λ such that P(λ) is singular. The coefficients of these terms are generated by sequences uk, vk of 1×n and n×1 vectors, respectively, which satisfy u1≠0, v1≠0, ∑k−1h=0(1⧸h!)uk−hP(h)(λ)=0, ∑k−1h=0(1⧸h!)P(h)(λ)vk−h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at λ but not necessarily a polynomial, the terms in the representation involving negative powers of z−λ provide the principal part of the Laurent expansion for P(z)−1 in a punctured neighborhood of z=λ