Matrix Analysis

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 Birkhoff’s Theorem.- II.3 Convex and Monotone Functions.- II.4 Binary Algebraic Operations and Majorisation.- II.5 Problems.- II.6 Notes and References.- III Variational Principles for Eigenvalues.- III.1 The Minimax Principle for Eigenvalues.- III.2 Weyl’s Inequalities.- III.3 Wielandt’s Minimax Principle.- III.4 Lidskii’s Theorems.- III. 5 Eigenvalues of Real Parts and Singular Values.- III.6 Problems.- III.7 Notes and References.- IV Symmetric Norms.- IV.l Norms on ?n.- IV.2 Unitarily Invariant Norms on Operators on ?n.- IV.3 Lidskii’s Theorem (Third Proof).- IV.4 Weakly Unitarily Invariant Norms.- IV.5 Problems.- IV.6 Notes and References.- V Operator Monotone and Operator Convex Functions.- V.1 Definitions and Simple Examples.- V.2 Some Characterisations.- V.3 Smoothness Properties.- V.4 Loewner’s Theorems.- V.5 Problems.- V.6 Notes and References.- VI Spectral Variation of Normal Matrices.- VI. 1 Continuity of Roots of Polynomials.- VI. 2 Hermitian and Skew-Hermitian Matrices.- VI. 3 Estimates in the Operator Norm.- VI. 4 Estimates in the Frobenius Norm.- VI. 5 Geometry and Spectral Variation: the Operator Norm.- VI. 6 Geometry and Spectral Variation: wui Norms.- VI. 7 Some Inequalities for the Determinant.- VI. 8 Problems.- VI. 9 Notes and References.- VII Perturbation of Spectral Subspaces of Normal Matrices.- VII. 1 Pairs of Subspaces.- VII. 2 The Equation AX — XB = Y.- VII. 3 Perturbation of Eigenspaces.- VII. 4 A Perturbation Bound for Eigenvalues.- VII.5 Perturbation of the Polar Factors.- VII. 6 Appendix: Evaluating the (Fourier) constants.- VII. 7 Problems.- VII. 8 Notes and References.- VIII Spectral Variation of Nonnormal Matrices.- VIII. 1 General Spectral Variation Bounds.- VIII. 4 Matrices with Real Eigenvalues.- VIII. 5 Eigenvalues with Symmetries.- VIII. 6 Problems.- VIII. 7 Notes and References.- IX A Selection of Matrix Inequalities.- IX. 1 Some Basic Lemmas.- IX. 2 Products of Positive Matrices.- IX. 3 Inequalities for the Exponential Function.- IX. 4 Arithmetic-Geometric Mean Inequalities.- IX. 5 Schwarz Inequalities.- IX. 6 The Lieb Concavity Theorem.- IX. 7 Operator Approximation.- IX. 8 Problems.- IX. 9 Notes and References.- X Perturbation of Matrix Functions.- X. 1 Operator Monotone Functions.- X. 2 The Absolute Value.- X. 3 Local Perturbation Bounds.- X. 4 Appendix: Differential Calculus.- X. 5 Problems.- X. 6 Notes and References.- References.