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E-grāmata: Totally Nonnegative Matrices

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Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics.

The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references.

Recenzijas

"This book is a very useful new reference on the subject of TN matrices and it will be of interest to researchers on matrix theory as well as to researchers of any field where total positivity has applications."---Juan Manuel Pefia, Mathematical Reviews

List of Figures
xi
Preface xiii
Chapter 0 Introduction
1(26)
0.0 Definitions and Notation
1(2)
0.1 Jacobi Matrices and Other Examples of TN matrices
3(12)
0.2 Applications and Motivation
15(9)
0.3 Organization and Particularities
24(3)
Chapter 1 Preliminary Results and Discussion
27(16)
1.0 Introduction
27(1)
1.1 The Cauchy-Binet Determinantal Formula
27(1)
1.2 Other Important Determinantal Identities
28(5)
1.3 Some Basic Facts
33(1)
1.4 TN and TP Preserving Linear Transformations
34(1)
1.5 Schur Complements
35(2)
1.6 Zero-Nonzero Patterns of TN Matrices
37(6)
Chapter 2 Bidiagonal Factorization
43(30)
2.0 Introduction
43(2)
2.1 Notation and Terms
45(2)
2.2 Standard Elementary Bidiagonal Factorization: Invertible Case
47(6)
2.3 Standard Elementary Bidiagonal Factorization: General Case
53(6)
2.4 LU Factorization: A consequence
59(3)
2.5 Applications
62(2)
2.6 Planar Diagrams and EB factorization
64(9)
Chapter 3 Recognition
73(14)
3.0 Introduction
73(1)
3.1 Sets of Positive Minors Sufficient for Total Positivity
74(6)
3.2 Application: TP Intervals
80(2)
3.3 Efficient Algorithm for testing for TN
82(5)
Chapter 4 Sign Variation of Vectors and TN Linear Transformations
87(10)
4.0 Introduction
87(1)
4.1 Notation and Terms
87(1)
4.2 Variation Diminution Results and EB Factorization
88(3)
4.3 Strong Variation Diminution for TP Matrices
91(3)
4.4 Converses to Variation Diminution
94(3)
Chapter 5 The Spectral Structure of TN Matrices
97(32)
5.0 Introduction
97(1)
5.1 Notation and Terms
98(1)
5.2 The Spectra of IITN Matrices
99(1)
5.3 Eigenvector Properties
100(6)
5.4 The Irreducible Case
106(12)
5.5 Other Spectral Results
118(11)
Chapter 6 Determinantal Inequalities for TN Matrices
129(24)
6.0 Introduction
129(2)
6.1 Definitions and Notation
131(1)
6.2 Sylvester Implies Koteljanskil
132(2)
6.3 Multiplicative Principal Minor Inequalities
134(12)
6.4 Some Non-principal Minor Inequalities
146(7)
Chapter 7 Row and Column Inclusion and the Distribution of Rank
153(14)
7.0 Introduction
153(1)
7.1 Row and Column Inclusion Results for TN Matrices
153(6)
7.2 Shadows and the Extension of Rank Deficiency in Submatrices of TN Matrices
159(6)
7.3 The Contiguous Rank Property
165(2)
Chapter 8 Hadamard Products and Powers of TN Matrices
167(18)
8.0 Definitions
167(1)
8.1 Conditions under which the Hadamard Product is TP/TN
168(1)
8.2 The Hadamard Core
169(8)
8.3 Oppenheim's Inequality
177(2)
8.4 Hadamard Powers of TP2
179(6)
Chapter 9 Extensions and Completions
185(20)
9.0 Line Insertion
185(1)
9.1 Completions and Partial TN Matrices
186(3)
9.2 Chordal Case-MLBC Graphs
189(2)
9.3 TN Completions: Adjacent Edge Conditions
191(4)
9.4 TN Completions: Single Entry Case
195(3)
9.5 TN Perturbations: The Case of Retractions
198(7)
Chapter 10 Other Related Topics on TN Matrices
205(14)
10.0 Introduction and Topics
205(1)
10.1 Powers and Roots of TP/TN Matrices
205(2)
10.2 Subdirect Sums of TN Matrices
207(5)
10.3 TP/TN Polynomial Matrices
212(1)
10.4 Perron Complements of TN Matrices
213(6)
Bibliography 219(20)
List of Symbols 239(6)
Index 245
Shaun M. Fallat is professor of mathematics and statistics at the University of Regina. Charles R. Johnson is the Class of 1961 Professor of Mathematics at the College of William & Mary.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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