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.
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