The QSVD or generalized SVD (GSVD) of
and
is
defined as follows.
Suppose
is
by
and
is
by
and nonsingular.
Let
.
Let the SVD of
. We may also write
two equivalent decompositions of
and
as
and
, where
is
by
and nonsingular,
is
by
and contains
in its leading
rows and columns and zeros elsewhere,
and
is
by
and contains
.
and
can be chosen so that
.
Thus
.
The diagonal entries
of
are called the generalized singular values of
and
.
This decomposition generalizes to the cases where
is singular or
by
,
to a decomposition equivalent to the SVD of
(the product SVD),
and indeed to decompositions of arbitrary products of the form
[108].