Indefinite Linear Algebra and Applications
This book is dedicated to relatively recent results in linear algebra of spaces with indefinite inner product. Within this framework, it presents the theory of subspaces and orthogonalization and then goes on to the theory of matrices, perturbation and stability theory. The book also includes applications of the theory to a study of matrix polynomials with selfadjoint constant coefficients, to differential and difference equations with constant coefficients, and to algebraic Ricatti equations.
After the notation and conventions, the book starts with basic geometric ideas concerning spaces with an indefinite inner product, the main topics here being orthogonalization, classification of subspaces and orthogonal polynomials. Further sections are devoted to a study of the classification of linear transformations in indefinite inner product spaces. H-selfadjoint, H-unitary and H-normal transformations together with their canonical forms are of particular interest. Functional calculus is discussed in the next chapter, where special attention is paid to the logarithmic and exponential functions. One chapter is used for a detailed analysis of the structure of H-normal matrices in spaces with an indefinite inner product. Following this, perturbation and stability theories for H-selfadjoint and H-unitary matrices are studied. This topic is important in applications involving the stable boundedness of solutions of differential and difference equations. One section is devoted to applications involving differential equations of the first order, the other for equations of higher orders. The last chapter contains the theory of algebraic Ricatti equations. The appendix serves as a refresher of some parts of linear algebra and matrix theory used in the main body of the book.
Each chapter ends with a series of examples that illustrates the discussed topics. The book has the structure of a graduate text in which chapters on advanced linear algebra form the core. This, together with many significant applications and an accessible style, makes it useful for engineers, scientists and mathematicians alike.