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Matrix Theory: Basic Results and Techniques Second Edition 2011 [Pehme köide]

  • Formaat: Paperback / softback, 399 pages, kõrgus x laius: 235x155 mm, kaal: 712 g, 1 Illustrations, color; 7 Illustrations, black and white, 1 Paperback / softback
  • Sari: Universitext
  • Ilmumisaeg: 31-Aug-2011
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1461410983
  • ISBN-13: 9781461410980
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  • Formaat: Paperback / softback, 399 pages, kõrgus x laius: 235x155 mm, kaal: 712 g, 1 Illustrations, color; 7 Illustrations, black and white, 1 Paperback / softback
  • Sari: Universitext
  • Ilmumisaeg: 31-Aug-2011
  • Kirjastus: Springer-Verlag New York Inc.
  • ISBN-10: 1461410983
  • ISBN-13: 9781461410980
Teised raamatud teemal:
The aim of this book is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. The book contains ten chapters covering various topics ranging from similarity and special types of matrices to Schur complements and matrix normality. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems.





Major changes in this revised and expanded second edition: -Expansion of topics such as matrix functions, nonnegative matrices, and (unitarily invariant) matrix norms -The inclusion of more than 1000 exercises; -A new chapter, Chapter 4, with updated material on numerical ranges and radii, matrix norms, and special operations such as the Kronecker and Hadamard products and compound matrices -A new chapter, Chapter 10, on matrix inequalities, which presents a variety of inequalities on the eigenvalues and singular values of matrices and unitarily invariant norms.





This book can be used as a textbook or a supplement for a linear algebra and matrix theory class or a seminar for senior undergraduate or graduate students. Prerequisites include a decent background in elementary linear algebra and calculus. The book can also serve as a reference for instructors and researchers in the fields of algebra, matrix analysis, operator theory, statistics, computer science, engineering, operations research, economics, and other fields.

Arvustused

From the reviews of the second edition:

This book provides an interesting and recommendable text for a second course on linear algebra with emphasis on the theory of matrices. (H. Mitsch, Monatshefte für Mathematik, Vol. 169 (1), January, 2013)

For the practitioner, someone new to the field, or users of matrices from other areas, there is a terse, but friendly, discussion of a number of useful topics, a few of which are not otherwise in book form. It is nice and friendly to read, with a simple, direct style. (Charles Johnson, SIAM Review, Vol. 55 (1), 2013)

The main purpose of this volume is to concisely present fundamental ideas, results, and techniques in linear algebra and mainly matrix theory. Each chapter focuses on the results, techniques, and methods that are beautiful, interesting, and representative, followed by carefully selected problems. The author has made a valuable contribution to the textbook literature on matrix theory, and his work will be appreciated by students and teachers of the subject. is recommended reading for all those wishing to acquaint themselves with basic matrix theory. (Viceniu D. Rdulescu, Zentralblatt MATH, Vol. 1229, 2012)

This interesting book is the revised version covering materials from the basic elements of matrix theory to more advanced topics. Each chapter starts with an introduction and includes a lot of well-selected problems . In many places several different proofs are presented for a theorem. This causes the book to be very attractive and readable. This book is useful for researchers as well as graduate students working in linear algebra, operator theory, statistics, computer science, engineering, applied mathematics, economics, and other disciplines. (Mohammad Sal Moslehian, Mathematical Reviews, Issue 2012 h)

Preface to the Second Edition vii
Preface ix
Frequently Used Notation and Terminology xv
Frequently Used Theorems xvii
1 Elementary Linear Algebra Review
1(34)
1.1 Vector Spaces
1(7)
1.2 Matrices and Determinants
8(9)
1.3 Linear Transformations and Eigenvalues
17(10)
1.4 Inner Product Spaces
27(8)
2 Partitioned Matrices, Rank, and Eigenvalues
35(38)
2.1 Elementary Operations of Partitioned Matrices
35(7)
2.2 The Determinant and Inverse of Partitioned Matrices
42(9)
2.3 The Rank of Product and Sum
51(6)
2.4 The Eigenvalues of AB and BA
57(5)
2.5 The Continuity Argument and Matrix Functions
62(5)
2.6 Localization of Eigenvalues: The Gersgorin Theorem
67(6)
3 Matrix Polynomials and Canonical Forms
73(34)
3.1 Commuting Matrices
73(6)
3.2 Matrix Decompositions
79(8)
3.3 Annihilating Polynomials of Matrices
87(6)
3.4 Jordan Canonical Forms
93(9)
3.5 The Matrices AT, A, A*, AT A, A*A, and AA
102(5)
4 Numerical Ranges, Matrix Norms, and Special Operations
107(18)
4.1 Numerical Range and Radius
107(6)
4.2 Matrix Norms
113(4)
4.3 The Kronecker and Hadamard Products
117(5)
4.4 Compound Matrices
122(3)
5 Special Types of Matrices
125(46)
5.1 Idempotence, Nilpotence, Involution, and Projections
125(8)
5.2 Tridiagonal Matrices
133(5)
5.3 Circulant Matrices
138(5)
5.4 Vandermonde Matrices
143(7)
5.5 Hadamard Matrices
150(5)
5.6 Permutation and Doubly Stochastic Matrices
155(9)
5.7 Nonnegative Matrices
164(7)
6 Unitary Matrices and Contractions
171(28)
6.1 Properties of Unitary Matrices
171(6)
6.2 Real Orthogonal Matrices
177(5)
6.3 Metric Space and Contractions
182(6)
6.4 Contractions and Unitary Matrices
188(4)
6.5 The Unitary Similarity of Real Matrices
192(3)
6.6 A Trace Inequality of Unitary Matrices
195(4)
7 Positive Semidefinite Matrices
199(54)
7.1 Positive Semidefinite Matrices
199(8)
7.2 A Pair of Positive Semidefinite Matrices
207(10)
7.3 Partitioned Positive Semidefinite Matrices
217(10)
7.4 Schur Complements and Determinant Inequalities
227(7)
7.5 The Kronecker and Hadamard Products of Positive Semidefinite Matrices
234(6)
7.6 Schur Complements and the Hadamard Product
240(5)
7.7 The Wielandt and Kantorovich Inequalities
245(8)
8 Hermitian Matrices
253(40)
8.1 Hermitian Matrices and Their Inertias
253(7)
8.2 The Product of Hermitian Matrices
260(6)
8.3 The Min-Max Theorem and Interlacing Theorem
266(8)
8.4 Eigenvalue and Singular Value Inequalities
274(7)
8.5 Eigenvalues of Hermitian matrices A, B, and A + B
281(6)
8.6 A Triangle Inequality for the Matrix (A*A)1/2
287(6)
9 Normal Matrices
293(32)
9.1 Equivalent Conditions
293(13)
9.2 Normal Matrices with Zero and One Entries
306(6)
9.3 Normality and Cauchy-Schwarz-Type Inequality
312(7)
9.4 Normal Matrix Perturbation
319(6)
10 Majorization and Matrix Inequalities
325(54)
10.1 Basic Properties of Majorization
325(9)
10.2 Majorization and Stochastic Matrices
334(6)
10.3 Majorization and Convex Functions
340(9)
10.4 Majorization of Diagonal Entries, Eigenvalues, and Singular Values
349(7)
10.5 Majorization for Matrix Sum
356(7)
10.6 Majorization for Matrix Product
363(9)
10.7 Majorization and Unitarily Invariant Norms
372(7)
References 379(12)
Notation 391(4)
Index 395
Fuzhen Zhang is currently a Professor of Mathematics at Nova Southeastern University.