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).