A pair of -dimensional subspaces
and
are called
(left and right) singular subspaces of
if
for all
and
for all
. We also write this as
and
.
The simplest example is when and
are spanned by
a single pair of singular vectors
and
of
, respectively.
More generally, any pair of singular subspaces can be spanned by a subset of
the singular vectors of
, although the spanning vectors do not have to
be singular vectors themselves.