**Student Mathematical Library**

Volume: 94;
2021;
217 pp;
Softcover

MSC: Primary 15; 26; 49;

**Print ISBN: 978-1-4704-6332-8
Product Code: STML/94**

List Price: $59.00

Individual Price: $47.20

**Electronic ISBN: 978-1-4704-6513-1
Product Code: STML/94.E**

List Price: $59.00

Individual Price: $47.20

#### Supplemental Materials

# Analysis and Linear Algebra: The Singular Value Decomposition and Applications

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*James Bisgard*

This book provides an elementary analytically
inclined journey to a fundamental result of linear algebra: the
Singular Value Decomposition (SVD). SVD is a workhorse in many
applications of linear algebra to data science. Four important
applications relevant to data science are considered throughout the
book: determining the subspace that “best” approximates a
given set (dimension reduction of a data set); finding the
“best” lower rank approximation of a given matrix
(compression and general approximation problems); the Moore-Penrose
pseudo-inverse (relevant to solving least squares problems); and the
orthogonal Procrustes problem (finding the orthogonal transformation
that most closely transforms a given collection to a given
configuration), as well as its orientation-preserving version.

The point of view throughout is analytic. Readers are assumed to
have had a rigorous introduction to sequences and continuity. These
are generalized and applied to linear algebraic ideas. Along the way
to the SVD, several important results relevant to a wide variety of
fields (including random matrices and spectral graph theory) are
explored: the Spectral Theorem; minimax characterizations of
eigenvalues; and eigenvalue inequalities. By combining analytic and
linear algebraic ideas, readers see seemingly disparate areas
interacting in beautiful and applicable ways.

#### Table of Contents

# Table of Contents

## Analysis and Linear Algebra: The Singular Value Decomposition and Applications

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Preface xi13
- Chapter 1. Introduction 121
- Chapter 2. Linear Algebra and Normed Vector Spaces 1333
- 2.1. Linear Algebra 1434
- 2.2. Norms and Inner Products on a Vector Space 2040
- 2.3. Topology on a Normed Vector Space 3050
- 2.4. Continuity 3858
- 2.5. Arbitrary Norms on ℝ^{𝕕} 4464
- 2.6. Finite-Dimensional Normed Vector Spaces 4868
- 2.7. Minimization: Coercivity and Continuity 5272
- 2.8. Uniqueness of Minimizers: Convexity 5474
- 2.9. Continuity of Linear Mappings 5676

- Chapter 3. Main Tools 6181
- Chapter 4. The Spectral Theorem 99119
- Chapter 5. The Singular Value Decomposition 123143
- Chapter 6. Applications Revisited 171191
- 6.1. The “Best” Subspace for Given Data 171191
- 6.2. Least Squares and Moore-Penrose Pseudo-Inverse 179199
- 6.3. Eckart-Young-Mirsky for the Operator Norm 182202
- 6.4. Eckart-Young-Mirsky for the Frobenius Norm and Image Compression 185205
- 6.5. The Orthogonal Procrustes Problem 188208
- 6.6. Summary 198218

- Chapter 7. A Glimpse Towards Infinite Dimensions 201221
- Bibliography 209229
- Index of Notation 213233
- Index 215235
- Back Cover Back Cover1239