Define as the
by
matrix whose top
rows contain
and whose bottom
rows are zero.
Define the
by
matrix
and
the
by
matrix
.
is called the left singular vector matrix of
,
and
is called the right singular vector matrix of
.
Since the
are orthogonal unit vectors, we see that
; i.e.,
is a unitary matrix. If
is real then the
are real vectors,
so
, and we also say that
is an orthogonal matrix.
The same discussion applies to
.
The
equalities
and
for
and
for
may also be written
and
, or
.
The factorization
There are several ``smaller'' versions of the SVD that are
often computed.
Let
be an
by
matrix
of the first
left singular vectors,
be an
by
matrix
of the first
right singular vectors,
and
be a
by
matrix of the first
singular values.
Then we can make the following definitions.
The thin SVD
may also be written
.
Each
is called a singular triplet.
The compact and truncated SVDs may be written similarly
(the sum going from
to
, or
to
, respectively).
If is
by
with
,
then its SVD is
, where
is
by
,
is
by
with
in its first
columns and
zeros in columns
through
, and
is
by
.
Its thin SVD is
, and the compact SVD and
truncated SVD are as above.
More generally, if we take a subset of columns of
and
(say
=
columns 2, 3, and 5, and
),
then these columns span a pair of singular subspaces of
.
If we take the corresponding submatrix
of
, then we can write the corresponding
partial SVD
. If the columns in
and
are replaced by
different orthonormal vectors spanning the same invariant subspace,
then we get a different partial SVD
,
where
is
a
by
matrix whose singular values are those of
, though
may not be diagonal.