Matrix Analysis [Paperback]

This book presents a substantial part of matrix analysis that is functional analytic in spirit. Topics covered include the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, and perturbation of matrix functions and matrix inequalities. The book offers several powerful methods and techniques of wide applicability, and it discusses connections with other areas of mathematics.

A good part of matrix theory is functional analytic in spirit. This statement can be turned around. There are many problems in operator theory, where most of the complexities and subtleties are present in the finite-dimensional case. My purpose in writing this book is to present a systematic treatment of methods that are useful in the study of such problems. This book is intended for use as a text for upper division and gradu? ate courses. Courses based on parts of the material have been given by me at the Indian Statistical Institute and at the University of Toronto (in collaboration with Chandler Davis). The book should also be useful as a reference for research workers in linear algebra, operator theory, mathe? matical physics and numerical analysis. A possible subtitle of this book could be Matrix Inequalities. A reader who works through the book should expect to become proficient in the art of deriving such inequalities. Other authors have compared this art to that of cutting diamonds. One first has to acquire hard tools and then learn how to use them delicately. The reader is expected to be very thoroughly familiar with basic lin? ear algebra. The standard texts Finite-Dimensional Vector Spaces by P.R.I A Review of Linear Algebra.- I.1 Vector Spaces and Inner Product Spaces.- I.2 Linear Operators and Matrices.- I.3 Direct Sums.- I.4 Tensor Products.- I.5 Symmetry Classes.- I.6 Problems.- I.7 Notes and References.- II Majorisation and Doubly Stochastic Matrices.- II.1 Basic Notions.- II. 2 Birkhoffs Theorem.- II.3 Convexl#.