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##

Singular Values and Singular Vectors

The square roots of the eigenvalues of are the
*singular values* of . Since is Hermitian and
positive semidefinite, the singular values are real and nonnegative. This lets
us write them in sorted order
.

The eigenvectors of are called *(right) singular vectors*.
We denote them by
, where is the eigenvector for
eigenvalue . The by matrix is also Hermitian
positive semidefinite. Its largest eigenvalues are identical
to those of , and the rest are zero. The eigenvectors of
are called *(left) singular vectors*. We denote them by
,
where through are eigenvectors for eigenvalues through
, and through are eigenvectors for the zero
eigenvalue. The singular vectors can be chosen to satisfy the
identities
and
for , and for
.

We may assume without loss of generality that each and
. The singular vectors are real if is real.
Though the singular vectors may not be unique (e.g., any vector
is a singular vector of the identity matrix), they may all be
chosen to be orthogonal to one another:
if .
When a singular value is distinct from all the other singular values,
its singular vectors are unique (up to multiplication by scalars).

** Next:** Singular Subspaces
** Up:** Singular Value Decomposition J.
** Previous:** Singular Value Decomposition J.
** Contents**
** Index**
Susan Blackford
2000-11-20