@article{LANGENHOP1977267,
    Abstract = {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−{$\lambda$} for each {$\lambda$} such that P({$\lambda$}) 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)({$\lambda$})=0, ∑k−1h=0(1⧸h!)P(h)({$\lambda$})vk−h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at {$\lambda$} but not necessarily a polynomial, the terms in the representation involving negative powers of z−{$\lambda$} provide the principal part of the Laurent expansion for P(z)−1 in a punctured neighborhood of z={$\lambda$}.},
    Author = {Langenhop, C.E.},
    File = {The Inverse of a Matrix Polynomial - 1-s2.0-002437957790009X-main - a - x.pdf},
    ISSN = {0024-3795},
    Journal = {Linear Algebra and its Applications},
    Number = {3},
    Pages = {267 - 284},
    Title = {The inverse of a matrix polynomial},
    URL = {http://www.sciencedirect.com/science/article/pii/002437957790009X},
    Volume = {16},
    Year = {1977},
    bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/002437957790009X},
    bdsk-url-2 = {https://doi.org/10.1016/0024-3795(77)90009-X},
    date-added = {2020-06-10 09:36:42 +0200},
    date-modified = {2020-06-10 09:36:42 +0200},
    doi = {10.1016/0024-3795(77)90009-X}
}

@article{LANGENHOP1977267, Abstract = {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−{$\lambda$} for each {$\lambda$} such that P({$\lambda$}) 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)({$\lambda$})=0, ∑k−1h=0(1⧸h!)P(h)({$\lambda$})vk−h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at {$\lambda$} but not necessarily a polynomial, the terms in the representation involving negative powers of z−{$\lambda$} provide the principal part of the Laurent expansion for P(z)−1 in a punctured neighborhood of z={$\lambda$}.}, Author = {Langenhop, C.E.}, File = {The Inverse of a Matrix Polynomial - 1-s2.0-002437957790009X-main - a - x.pdf}, ISSN = {0024-3795}, Journal = {Linear Algebra and its Applications}, Number = {3}, Pages = {267 - 284}, Title = {The inverse of a matrix polynomial}, URL = {http://www.sciencedirect.com/science/article/pii/002437957790009X}, Volume = {16}, Year = {1977}, bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/002437957790009X}, bdsk-url-2 = {https://doi.org/10.1016/0024-3795(77)90009-X}, date-added = {2020-06-10 09:36:42 +0200}, date-modified = {2020-06-10 09:36:42 +0200}, doi = {10.1016/0024-3795(77)90009-X} }