Analysis and linear algebra : the singular value decomposition and applications
著者
書誌事項
Analysis and linear algebra : the singular value decomposition and applications
（Student mathematical library, v. 94）
American Mathematical Society, c2021
- : pbk
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注記
IncludIncludes bibliographical references (p. 209-212) and indexes
内容説明・目次
内容説明
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.
目次
Introduction
Linear algebra and normed vector spaces
Main tools
The spectral theorem
The singular value decomposition
Applications revisited
A glimpse towards infinite dimensions
Bibliography
Index of notation
Index
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