Analysis and Linear Algebra: The Singular Value Decomposition and Applications

✍ Scribed by James Bisgard (author)

• Publisher: American Mathematical Society
• Year: 2021
• Tongue: English
• Leaves: 239
• Series: Student Mathematical Library 94
• Category: Library

✦ Synopsis

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✦ Table of Contents

Cover
Title page
Copyright
Contents
Preface
Pre-Requisites
Notation
Acknowledgements
Chapter 1. Introduction
1.1. Why Does Everybody Say Linear Algebra is “Useful”?
1.2. Graphs and Matrices
1.3. Images
1.4. Data
1.5. Four “Useful” Applications
Chapter 2. Linear Algebra and Normed Vector Spaces
2.1. Linear Algebra
2.2. Norms and Inner Products on a Vector Space
2.3. Topology on a Normed Vector Space
2.4. Continuity
2.5. Arbitrary Norms on ℝ^{𝕕}
2.6. Finite-Dimensional Normed Vector Spaces
2.7. Minimization: Coercivity and Continuity
2.8. Uniqueness of Minimizers: Convexity
2.9. Continuity of Linear Mappings
Chapter 3. Main Tools
3.1. Orthogonal Sets
3.2. Projection onto (Closed) Subspaces
3.3. Separation of Convex Sets
3.4. Orthogonal Complements
3.5. The Riesz Representation Theorem and Adjoint Operators
3.6. Range and Null Spaces of 𝐿 and 𝐿*
3.7. Four Problems, Revisited
Chapter 4. The Spectral Theorem
4.1. The Spectral Theorem
4.2. Courant-Fischer-Weyl Min-Max Theorem for Eigenvalues
4.3. Weyl’s Inequalities for Eigenvalues
4.4. Eigenvalue Interlacing
4.5. Summary
Chapter 5. The Singular Value Decomposition
5.1. The Singular Value Decomposition
5.2. Alternative Characterizations of Singular Values
5.3. Inequalities for Singular Values
5.4. Some Applications to the Topology of Matrices
5.5. Summary
Chapter 6. Applications Revisited
6.1. The “Best” Subspace for Given Data
6.2. Least Squares and Moore-Penrose Pseudo-Inverse
6.3. Eckart-Young-Mirsky for the Operator Norm
6.4. Eckart-Young-Mirsky for the Frobenius Norm and Image Compression
6.5. The Orthogonal Procrustes Problem
6.6. Summary
Chapter 7. A Glimpse Towards Infinite Dimensions
Bibliography
Index of Notation
Index
Back Cover