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
Share this pageJames 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