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: The book is appropriate for 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.
Contents: Preface to the Classics Edition; Preface; Errata; Introduction; Part I: Monic Matrix Polynomials: Chapter 1: Linearization and Standard Pairs; Chapter 2: Representation of Monic Matrix Polynomials; Chapter 3: Multiplication and Divisability; Chapter 4: Spectral Divisors and Canonical Factorization; Chapter 5: Perturbation and Stability of Divisors; Chapter 6: Extension Problems; Part II: Nonmonic Matrix Polynomials: Chapter 7: Spectral Properties and Representations; Chapter 8: Applications to Differential and Difference Equations; Chapter 9: Least Common Multiples and Greatest Common Divisors of Matrix Polynomials; Part III: Self-Adjoint Matrix Polynomials: Chapter 10: General Theory; Chapter 11: Factorization of Self-Adjoint Matrix Polynomials; Chapter 12: Further Analysis of the Sign Characteristic; Chapter 13: Quadratic Self-Adjoint Polynomials; Part IV: Supplementary Chapters in Linear Algebra: Chapter S1: The Smith Form and Related Problems; Chapter S2: The Matrix Equation AX XB = C; Chapter S3: One-Sided and Generalized Inverses; Chapter S4: Stable Invariant Subspaces; Chapter S5: Indefinite Scalar Product Spaces; Chapter S6: Analytic Matrix Functions; References; List of Notation and Conventions; Index