Unitary variational formula for the trace norm
= Unitary variational formula for the trace norm
{title2=$\|X\|_1=\max_U|\operatorname{Tr}(XU)|$}
For a finite square matrix $X$, maximize over <unitary operators>. Write its <singular value decomposition> as $X=PDQ^\dagger$. Then $|\operatorname{Tr}(XU)|=|\operatorname{Tr}(DQ^\dagger UP)|\leq\sum_jD_{jj}$, since every diagonal entry of a unitary has modulus at most one. Choosing $U=QP^\dagger$ attains equality, proving the formula for the <trace norm>.