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Topics in Matrix Analysis

Book Description

Building on the foundations of its predecessor volume, Matrix Analysis, this book treats in detail several topics in matrix theory not included in the previous volume, but with important applications and of special mathematical interest. As with the previous volume, the authors assume a background knowledge of elementary linear algebra and rudimentary analytical concepts. Many examples and exercises of varying difficulty are included.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Preface
  6. Chapter 1 The field of values
    1. 1.0 Introduction
    2. 1.1 Definitions
    3. 1.2 Basic properties of the field of values
    4. 1.3 Convexity
    5. 1.4 Axiomatization
    6. 1.5 Location of the field of values
    7. 1.6 Geometry
    8. 1.7 Products of matrices
    9. 1.8 Generalizations of the field of values
  7. Chapter 2 Stable matrices and inertia
    1. 2.0 Motivation
    2. 2.1 Definitions and elementary observations
    3. 2.2 Lyapunov’s theorem
    4. 2.3 The Routh-Hurwitz conditions
    5. 2.4 Generalizations of Lyapunov’s theorem
    6. 2.5 M-matrices, P-matrices, and related topics
  8. Chapter 3 Singular value inequalities
    1. 3.0 Introduction and historical remarks
    2. 3.1 The singular value decomposition
    3. 3.2 Weak majorization and doubly substochastic matrices
    4. 3.3 Basic inequalities for singular values and eigenvalues
    5. 3.4 Sums of singular values: the Ky Fan k-norms
    6. 3.5 Singular values and unitarily invariant norms
    7. 3.6 Sufficiency of Weyl’s product inequalities
    8. 3.7 Inclusion intervals for singular values
    9. 3.8 Singular value weak majorization for bilinear products
  9. Chapter 4 Matrix equations and the Kronecker product
    1. 4.0 Motivation
    2. 4.1 Matrix equations
    3. 4.2 The Kronecker product
    4. 4.3 Linear matrix equations and Kronecker products
    5. 4.4 Kronecker sums and the equation AX + XB = C
    6. 4.5 Additive and multiplicative commutators and linear preservers
  10. Chapter 5 The Hadamard product
    1. 5.0 Introduction
    2. 5.1 Some basic observations
    3. 5.2 The Schur product theorem
    4. 5.3 Generalizations of the Schur product theorem
    5. 5.4 The matrices A ∘ (A[sup(-1)])[sup(T)] and A ∘ A(A[sup(-1)])
    6. 5.5 Inequalities for Hadamard products of general matrices: an overview
    7. 5.6 Singular values of a Hadamard product: a fundamental inequality
    8. 5.7 Hadamard products involving nonnegative matrices and M-matrices
  11. Chapter 6 Matrices and functions
    1. 6.0 Introduction
    2. 6.1 Polynomial matrix functions and interpolation
    3. 6.2 Nonpolynomial matrix functions
    4. 6.3 Hadamard matrix functions
    5. 6.4 Square roots, logarithms, nonlinear matrix equations
    6. 6.5 Matrices of functions
    7. 6.6 A chain rule for functions of a matrix
  12. Hints for problems
  13. References
  14. Notation
  15. Index