Schmidt decomposition

Choose ordered orthonormal bases $(e_i{)}_{i=1}^m$ of $H$ and $(f_k{)}_{k=1}^n$ of $K$ using [F4], and write $\mathrm{\Psi }={\sum }_{i,k}{c}_{ik}{e}_i\otimes f_k$ by [F1]. Define the complex-linear map $T_C:{\mathbb{C}}^n\to {\mathbb{C}}^m$ by $(T_{C}z{)}_i={\sum }_{k=1}^n{c}_{ik}{z}_k$; its standard matrix is exactly $C$. Apply the positive-family construction [F3] to $T_C$. If $q_j$ and $p_j$ are its right and positive left singular vectors, the identity there gives $T_{C}z={\sum }_{j=1}^r{s}_j⟨z,q_j⟩p_j$. Evaluating at each standard basis vector of ${\mathbb{C}}^n$ and using [F5], the $(i,k)$ entry of this identity is $c_{ik}={\sum }_{j=1}^r{s}_j(p_j{)}_i\overline{(q_j{)}_k}$. This is the coefficient formula needed below, obtained from the positive singular families without a codomain basis completion.

Define $u_j={\sum }_i(p_j{)}_i{e}_i$ and $v_j={\sum }_k\overline{(q_j{)}_k}{f}_k$. The coordinate columns of the $p_j$ are orthonormal by [F3]–[F5], so the $u_j$ are orthonormal. For the second family, $⟨v_j,v_{\mathrm{\ell }}⟩={\sum }_k\overline{(q_j{)}_k}(q_{\mathrm{\ell }}{)}_k=\overline{⟨q_j,q_{\mathrm{\ell }}⟩}={\delta }_{j\mathrm{\ell }}$, so it too is orthonormal. The coefficient of $e_i\otimes f_k$ in ${\sum }_j{s}_j{u}_j\otimes v_j$ is ${\sum }_j{s}_j(p_j{)}_i\overline{(q_j{)}_k}=c_{ik}$ by step 1.1, so this sum equals $\mathrm{\Psi }$. If $C=0$, then $\mathrm{\Psi }=0$ and $r=0$. The positive coefficients are precisely the positive singular values of $C$.

By [F1], the vectors $u_j\otimes v_j$ are orthonormal: their pairings are $⟨u_i,u_j⟩⟨v_i,v_j⟩={\delta }_{ij}$. Expanding the squared norm of the sum in step 2.1 therefore gives $\mathrm{\parallel }\mathrm{\Psi }{\mathrm{\parallel }}^2={\sum }_j{s}_j^2$, including the empty sum when $\mathrm{\Psi }=0$.