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Matrix Analysis [electronic resource] / by Rajendra Bhatia.
Uniform title:
New York, NY : Springer New York : Imprint: Springer, 1997.
XI, 349 p. online resource.
Graduate Texts in Mathematics, 0072-5285 ; 169
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
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