Author | Bhatia, Rajendra. author |
---|---|
Title | Matrix Analysis [electronic resource] / by Rajendra Bhatia |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1997 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0653-8 |
Descript | XI, 349 p. online resource |
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 Birkhoffโs 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 Weylโs Inequalities -- III.3 Wielandtโs Minimax Principle -- III.4 Lidskiiโs 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 Lidskiiโs 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 Loewnerโs 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 โ 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