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This book is the definitive 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. 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.
Preface to the Classics Edition; Preface; Errata; Introduction; Part I. Monic Matrix Polynomials: 1. Linearization and standard pairs; 2. Representation of monic matrix polynomials; 3. Multiplication and divisibility; 4. Spectral divisors and canonical factorization; 5. Perturbation and stability of divisors; 6. Extension problems; Part II. Nonmonic Matrix Polynomials: 7. Spectral properties and representations; 8. Applications to differential and difference equations; 9. Least common multiples and greatest common divisors of matrix polynomials; Part III. Self-Adjoint Matrix Polynomials: 10. General theory; 11. Factorization of self-adjoint matrix polynomials; 12. Further analysis of the sign characteristic; 13: Quadratic self-adjoint polynomials; Part IV. Supplementary Chapters in Linear Algebra: S1. The Smith form and related problems; S2. The matrix equation AX – XB = C; S3. One-sided and generalized inverses; S4. Stable invariant subspaces; S5. Indefinite scalar product spaces; S6. Analytic matrix functions; References; List of notation and conventions; Index.
Overview
This book is the definitive 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 ...