Lecture The Singular Value
Decomp osition
The singular value decomp osition SVD is a matrix factorization whose com
putation is a step in many algorithms Equally imp ortant is the use of the
SVD for conceptual purp oses Many problems of linear algebra can b e b etter
ask the question what if we take the SVD understo o d if we rst
A Geometric Observation
The SVD is motivated by the following geometric fact
The image of the unit sphere under any m n matrix is a hyperel lipse
The SVD is applicable to b oth real and complex matrices However in de
scribing the geometric interpretation we assume as usual that the matrix is
real
yp erellipse may b e unfamiliar but this is just the m dimen The term h
m
sional generalization of an ellipse We may de ne a hyp erellipse in IR as
m
the surface obtained by stretching the unit sphere in IR by some factors
m