schatten_norm_le_rank_rpow_smul_spectral_norm

For a real $n_1\times n_2$ matrix $X$ and any exponent $q\ge 1$, the Schatten-$q$ norm is controlled by the rank and the spectral norm: $\lVert X\rVert_{S_q}\le r^{1/q}\,\lVert X\rVert$, where $r=\operatorname{rank}(X)$ (Module.finrank ℝ (LinearMap.range (Matrix.toEuclideanLin X))), $\lVert X\rVert_{S_q}=\bigl(\sum_k\sigma_k(X)^q\bigr)^{1/q}$ is schattenNorm q X, and $\lVert X\rVert=\sigma_0(X)$ is spectralNorm X. The bound holds because every singular value satisfies $\sigma_k\le\sigma_0=\lVert X\rVert$ and only the first $r$ singular values are nonzero, so $\sum_k\sigma_k^q\le r\,\sigma_0^q$ and hence $\bigl(\sum_k\sigma_k^q\bigr)^{1/q}\le r^{1/q}\,\sigma_0$. With $q\ge\log r$ the prefactor $r^{1/q}$ collapses to $\le e^{1/2}$ (the window-collapse handled separately by Node-B / gram_schatten_le_exp_half_variance_scale). Source: Candès–Recht 2009 (arXiv:0805.4471), §6.1, the operator/Schatten comparison right after Lemma 6.1; Horn–Johnson §5.6, §7.3.