tangent_coordinate_kernel_offdiagonal_bound_from_a0_min_dim

Off-diagonal tangent-kernel magnitude bound (Candes–Recht 2009, §6 eq (6.1)+(6.2), p.32). For a rank-$r$ SVD $S$ of $M\in\mathbb R^{n_1\times n_2}$ satisfying the standard incoherence hypothesis A0 S μ₀ (the $A_0$ coherence bound with parameter $\mu_0\ge 1$), the tangent-space coordinate kernel $K(i,j,a,b)=\langle P_T(e_ie_j^\top),\,e_ae_b^\top\rangle$ is uniformly bounded — for all index pairs $(i,j),(a,b)$, not only the diagonal $(i,j)=(a,b)$ — by

This is the off-diagonal generalization of the diagonal estimate (4.8) $\|P_T(e_ae_b^\top)\|_F^2\le 2\mu_0 r/\min(n_1,n_2)$. It follows by Cauchy–Schwarz for the Frobenius inner product: $K(i,j,a,b)=\langle P_T e_{ij},P_T e_{ab}\rangle$ (self-adjointness and idempotence of $P_T$), hence $|K(i,j,a,b)|\le\|P_T e_{ij}\|_F\,\|P_T e_{ab}\|_F$, and each factor equals $\sqrt{K(\cdot,\cdot,\cdot,\cdot)_{\text{diag}}}\le\sqrt{C_{\mathrm{ker}}\mu_0 r/\min}$ by eq (4.7).