Matrix Theory : Basic Results and Techniques.

Format:

Book

Author/Creator:

Zhang, Fuzhen.

Series:

Universitext Series

Language:

English

Physical Description:

1 online resource (513 pages)

Edition:

3rd ed.

Place of Publication:

New York, NY : Springer, 2026.

Summary:

The aim of this text is to present fundamental ideas, results, and techniques concisely, mainly in matrix theory with some in linear algebra.The book contains ten chapters covering various topics ranging from rank, similarity, and special matrices, to Schur complements, matrix normality, and majorization.

Contents:

Intro

Preface to the Third Edition

Contents

Frequently Used Notation and Terminology

Frequently Used Theorems

CHAPTER 1 Elementary Linear Algebra Review

1.1 Vector Spaces

1.2 Matrices and Determinants

1.3 Linear Transformations, Matrices, and Their Eigenvalues

1.4 Inner Product Spaces

CHAPTER 2 Partitioned Matrices, Ranks, and Eigenvalues

2.1 Partitioned Matrices and Their Elementary Operations

2.2 Determinants and Inverses of Partitioned Matrices

2.3 The Ranks of Product AB and Sum A + B

2.4 The Eigenvalues of AB, BA, and 0A∗A0

2.5 Localization of Eigenvalues: The Gerˇsgorin Disc Theorem

2.6 The Continuity Argument and Matrix Functions

CHAPTER 3 Matrix Polynomials and Canonical Forms

3.1 Commuting Matrices

3.2 Matrix Decompositions

3.3 Annihilating Polynomials of Matrices

3.4 λ-matrices and the Jordan Canonical Form

3.5 The Matrices AT, A, A∗, ATA, A∗A, and AA

CHAPTER 4 Numerical Ranges, Norms, and Special Products of Matrices

4.1 Numerical Range and Numerical Radius

4.2 Matrix Norms

4.3 The Kronecker and Hadamard Products

4.4 Compound Matrices

CHAPTER 5 Special Types of Matrices

5.1 Idempotence, Nilpotence, Involution, and Projections

5.2 Tridiagonal Matrices

5.3 Circulant Matrices

5.4 Vandermonde Matrices

5.5 Hadamard Matrices

5.6 Permutation Matrices and Doubly Stochastic Matrices

5.7 Nonnegative Matrices

CHAPTER 6 Unitary Matrices and Contractive Matrices

6.1 Properties of Unitary Matrices

6.2 Real Orthogonal Matrices

6.3 Contractive Matrices

6.4 The Unitary Similarity of Real Matrices

6.5 An Inequality of Unit Vectors and Its Applications

CHAPTER 7 Positive Semidefinite Matrices

7.1 Positive Semidefinite Matrices

7.2 A Pair of Positive Semidefinite Matrices

7.3 Partitioned Matrices and The Schur Complement.

7.4 The Schur Complement and Determinant Inequalities

7.5 The Kronecker and Hadamard Products and Inequalities

7.6 Inequalities of Choi, Kadison, Kantorovich, and Wielandt

CHAPTER 8 Hermitian Matrices

8.1 Hermitian Matrices and Their Inertias

8.2 The Product of Hermitian Matrices

8.3 The Min-max Principles and Interlacing Theorems

8.4 Eigenvalue and Singular Value Inequalities

8.5 Sum of Eigenvalues of Hermitian Matrices A,B, and A + B

8.6 A Triangle Inequality for the Matrix |A| = (A∗A)1/2

CHAPTER 9 Normal Matrices

9.1 Equivalent Conditions

9.2 Normal Matrices with Zero and One Entries

9.3 Normality and Cauchy-Schwarz Type Inequalities

9.4 Spectral Perturbation of Normal Matrices

CHAPTER 10 Majorization and Matrix Inequalities

10.1 Basic Properties of Majorization

10.2 Majorization and Stochastic Matrices

10.3 Majorization and Convex Functions

10.4 Majorization of Diagonal Entries, Eigenvalues, and Singular Values

10.5 Majorization for Matrix Sum

10.6 Majorization for Matrix Product

10.7 Majorization and Unitarily Invariant Norms

References for the 3rd Edition

Notation

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

1-0716-5238-9

9781071652381