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.