Multiparameter Eigenvalue Problems

Name: Multiparameter Eigenvalue Problems
Brand: Academic Press
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ATKINSON(Author)

ATKINSON(Editor)

Academic Press

Published on 14. May 2014

225 pages

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978-0-08-095590-2 (ISBN)

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Front Cover
Multiparameter Eigenvalue Problems
Copyright Page
Contents
Preface
Contents of Volume II
PART I: PRELIMINARIES FROM LINEAR ALGEBRA
Chapter 1. Linear Spaces
1.1 Introduction
1.2 Linear Maps
1 3 Composite and Induced Maps
1.4 Direct Sums
1.5 Linear Dependence and Dimension
1.6 Dimensions of Kernel and Image
1.7 Further Dimensional Results
1.8 Topologies
1.9 Connectedness
1.10 Semilinear Maps
Chapter 2. Bilinear and Multilinear Functions
2.1 Multilinear Functions
2.2 Bilinear Functions
2.3 Bilinear Functions on a Single Space
2.4 Bilinear Forms
2.5 Sesquilinear Functions
2.6 Sesquilinear Forms and Endoniorphisms
2.7 The Zeros of Hermitian Forms
2.8 Pairs of Hermitian Forms
2.9 Three Hermitian Forms
2.10 General Remarks on the Range of a Set of Forms
Chapter 3. The Decomposition of Finite-Dimensional Endomorphisms
3.1 Ascent and Descent
3.2 The Case of Equal Ascent and Descent
3.3 Eigensubspaces and Root Subspaces
3.4 The Splitting Off of Root Subspaces
3.5 The Finite-Dimensional Case
3.6 Several Commuting Operators
3.7 The Hermitian Case
3.8 Orthogonality
3.9 Some Modifications
3.10 Reduction of Pairs of Hermitian Forms
Chapter 4. The Tensor Product of Linear Spaces
4.1 Introduction
4.2 The Definition by Means of Functionals
4.3 Bases and Dimension
4.4 The Real and Complex Cases
4.5 Subspaces
4.6 Induced Homomorphisms
4.7 Exactness Properties
4.8 Universal Property
4.9 Bilinear Forms and Tensor Products
4.10 Products of Sesquilinear Forms
Chapter 5. Tensor Products and Endomorphisms
5.1 Introduction
5.2 The Hermitian Case
5.3 Eigenvalues and Ranks
5.4 Decomposition
5.5 The Kronecker Sum and Product
5.6 Kronecker Sums and Eigenvalues
5.7 Separation of Variables
5.8 The Tensor Product of Identical Factors
5.9 Induced Maps of Symmetry Subspaces
PART 2: MULTIPARAMETER PROBLEMS FOR MATRICES
Chapter 6. Simultaneous Equations in Linear Spaces
6.1 Introduction
6.2 Determinantal Maps
6.3 Singular Determinantal Maps in the Case k = 2
6.4 Rectangular Arrays
6.5 Definiteness Requirements
6.6 Solutions for Rectangular Arrays
6.7 Nonformal Determinantal Properties
6.8 Eigenvalues for a Rectangular Array
6.9 Decomposition
Chapter 7. Simultaneous Eigenvalue Problems for Hermitian Matrices
7.1 Introduction
7.2 The First Definiteness Condition and Its Consequences
7.3 Orthogonality of Eigenvectors
7.4 Stronger Definiteness Conditions
7.5 Splitting of Multiple Eigenvalues
7.6 Decomposable Orthogonal Eigenvectors
7.7 A Connectedness Property
7.8 The Main Result on Positive Definiteness
7.9 The Eigenvector Expansion
Chapter 8. The Singularity of Square Arrays
8.1 Introduction
8.2 Equivalent Singularity Conditions
8.3 An Algebraic Lemma
8.4 The Inductive Argument
8.5 Singularity and Decomposable Tensors
8.6 The Positive Definiteness Theorem Reproved
8.7 Eigenvalues and Singularity
Chapter 9. Arrays of Hermitian Forms
9.1 Introduction
9.2 Two by Two Arrays
9.3 Two by Three Arrays
9.4 General Square Arrays
9.5 A Property of Convex Cones
9.6 The Case of Several Cones
9.7 Square Arrays of Hermitian Forms, Continued
9.8 Rectangular Arrays of Hermitian Forms
9.9 Relation between Definiteness Conditions I and II
Chapter 10. Completeness of Eigenvectors in the Locally Definite Case
10.1 Introduction
10.2 Eigenvalues and Eigensubspaces
10.3 Eigenprojectors
10.4 Existence of a Nonsingular Determinantal Map
10.5 Completeness of the Eigenvectors
10.6 The Eigenvector Expansion
Chapter 11. Arrays of Compact Operators
11.1 Introduction
11.2 Notions from Hilbert Space Theory
11.3 Discreteness of the Spectrum
11.4 Truncated Problems
11.5 Sequences of Truncations
11.6 Convergence of the Eigenvalues
11.7 Convergence of the Eigenvectors
11.8 Introduction of Tensor Products
11.9 Discussion of the Expansion
11.10 A Special Case
References
Index

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