nuclear_norm_dual_achiever_contraction

Nuclear-norm dual achiever (Candès–Recht 2009, arXiv:0805.4471, Lemma 3.2, p.15). For every real matrix $N$ there is a contraction $Z$ in the operator (spectral) norm, $\lVert Z\rVert\le 1$, that realizes the nuclear norm of $N$ as a Frobenius inner product: $\langle Z, N\rangle = \lVert N\rVert_*$. Concretely, if $N=\sum_\ell\tilde\sigma_\ell\tilde u_\ell\tilde v_\ell^\top$ is an SVD of $N$, then $Z=\operatorname{sign}(N)=\sum_\ell\tilde u_\ell\tilde v_\ell^\top$ is a partial isometry with $\lVert Z\rVert=1$ (or $0$ when $N=0$) and $\langle Z,N\rangle=\sum_\ell\tilde\sigma_\ell=\lVert N\rVert_*$. This is the dual-norm duality $\lVert N\rVert_*=\max_{\lVert Z\rVert\le 1}\langle Z,N\rangle$, achieved at the sign matrix; it is the achiever half of trace duality (the equality case complementing the von Neumann inequality $\langle Z,N\rangle\le\lVert Z\rVert\,\lVert N\rVert_*$).