The inverse of a matrix polynomial

An 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 u_k, v_k of 1×n and n×1 vectors, respectively, which satisfy u₁≠0, v₁≠0, ∑^k?1_h=0(1?h!)u_k?hP^(h)(λ)=0, ∑^k?1_h=0(1?h!)P^(h)(λ)v_k?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.