Matrix Analysis [electronic resource] / by Rajendra Bhatia.

Author(s):

Uniform title:

Imprint:

New York, NY : Springer New York : Imprint: Springer, 1997.

Description:

XI, 349 p. online resource.

ISBN:

9781461206538

Subject(s):

Series:

Graduate Texts in Mathematics, 0072-5285 ; 169

Summary:

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.

Contents

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.

Journal citation:

Springer eBooks

Record created 2017-03-01, last modified 2018-03-22