The trace norm (or nuclear norm) of a matrix is the sum of its singular values:
We show for a matrix , its trace norm is equal to:
in other words, the dual norm of trace norm is spectral norm .
Let the SVD of and
to be
and
respectively.
where and
are singular values of
. The equality holds when
and
are non-negative for all i.
Let be its SVD. We have
, where
and
are singular values.
Apply Cauchy-Schwarz inequality:
With Lemma 1, we know and equality holds when
.
Let be its SVD. For any
, we have
.
Let and
. Because
and
are unitary matrices,
and
. Therefore:
The equality holds when and
.