Suppose and
are unitary matrices, i.e.,
and
.
If
and
are real, then
and
,
and we call them
orthogonal matrices. Let
.
We say that
is
unitarily (orthogonally) equivalent to
and that
and
are unitary (orthogonal)
equivalence transformations.
has the same singular values as
.
If
and
are left and right singular vectors of
, respectively,
so that
and
, then
and
are
left and right singular vectors of
, respectively.