An introduction to applied matrix analysis (Q2813000)

This textbook is intended for a one-semester course in applied matrix theory. It starts with a preliminary chapter which introduces the basic symbols, notations and results used: quadratic forms, positive definite matrices, eigenvalues and eigenvectors, Kronecker product and sum. Chapter 2 presents results on vector and matrix norms, as well as floating point representation and perturbation analysis. In Chapter 3, the authors study the linear least square problem (LLSP) for overdetermined matrices, together with the QR algorithm for its numerical solution. Chapter 4 presents the Moore-Penrose generalized inverse and its connection with LLSP. In Chapter 5, the authors present the steepest descent and conjugate gradient algorithms, together with some preliminary considerations on preconditioning techniques. Chapter 6 and 7 develop the analysis of various optimal and superoptimal preconditioning methods. In the last Chapter 8, the authors present a proof for the Böttcher-Wenzel conjecture and related matrix inequalities. The textbook is accessible to students with a background in linear algebra, calculus and numerical analysis, but also to researchers from other fields interested in solving linear systems of equations.