Linear Algebra in Action
This is a book on linear algebra, playing an important role in pure and applied mathematics, computer science, physics and engineering. The book is divided into 23 chapters and 2 appendices. The first six chapters and some selected parts from chapters 7-9 are based on classical linear algebra topics. The reader will find here many interesting principles and results concerning vector spaces, Gaussian elimination and its applications, determinants, eigenvalues and eigenvectors, Jordan forms and their calculations, normed linear spaces, inner product spaces and orthogonality, and symmetric, Hermitian and normal matrices. The next three chapters are devoted to singular values and related inequalities, pseudoinverses and triangular factorization and positive definite matrices.
Chapter 13 treats difference equations, differential equations and their systems. Chapters 14-16 contain applications to vector valued functions, the implicit function theorem and extremal problems. The subsequent chapters deal with matrix valued holomorphic functions, matrix equations, realization theory, eigenvalue location problems, zero location problems, convexity and matrices with nonnegative entries. Two appendices describe useful facts from complex function theory. The book offers basic and advanced techniques of linear algebra from the point of view of analysis. Each technique is illustrated by a wide sample of applications and it is accompanied by many exercises of varying difficulty, which give further extensions of the theory. The book can be recommended as a general text for a variety of courses on linear algebra and its applications, as well as a self-study aid for graduate and undergraduate students.