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. Language: English