This book provides a comprehensive treatment of the theory of polynomials in a complex variable with matrix coefficients. Basic matrix theory can be viewed as the study of the special case of polynomials of first degree; the theory developed in Matrix Polynomials is a natural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations, systems theory, the Wiener-Hopf technique, mechanics and vibrations, and numerical analysis. Although there have been significant advances in some quarters, this work remains the only systematic development of the theory of matrix polynomials. Audience: students, instructors, and researchers in linear algebra, operator theory, differential equations, systems theory, and numerical analysis. Its contents are accessible to readers who have had undergraduate-level courses in linear algebra and complex analysis.
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admissible pair canonical factorization canonical set Chapter col(X columns comonic matrix polynomial companion matrix complex numbers Consider Corollary corresponding decomposable pair deﬁned deﬁnition denote eigenvalue eigenvector elementary divisors equality exists F-spectral ﬁnd ﬁnite Jordan pair ﬁrst ﬁxed follows formula given Gohberg greatest common divisor implies ind(X inﬁnite integer invariant subspace invertible invertible matrix Jordan block Jordan chains Jordan matrix L(ll left inverse Lemma linear transformation linearly independent matrix function matrix poly matrix polynomial L(/l monic divisor monic matrix polynomial monic polynomials monic right divisor nomials nonsingular matrix nonzero obtain operator polynomials partial multiplicities polynomial L(1 polynomial of degree problem projector proof of Theorem Proposition prove rank real eigenvalues resp satisﬁed scalar polynomial Section self-adjoint matrix polynomial self-adjoint triple set of Jordan sign characteristic Smith form solution standard pair standard triple supporting subspace Theorem theory triple X unique vectors zero