I'm trying to understand the proof of the Schmidt decomposition found on the Wikipedia page. In particular, the part that I've circled in red in this screenshot:
I've tried writing out $U_1, \Sigma, V$ with general coordinates $u_{i,j}, \sigma_{i,j}, v_{i,j}$, and multiplying things out, but I just find a way to manipulate it into the single sum over $k$.
Can someone show me how to obtain the red-circled equation from the line above it?
$$(M_w)_{i,j} = \sum\limits_{k,l = 1}^m u_{i,k}\sigma_{k,l}\bar{v}_{j,l} = \sum\limits_{k,l = 1}^m u_{i,k}\delta_{k,l}\alpha_k\bar{v}_{j,l} = \sum\limits_{k = 1}^m \alpha_k u_{i,k}\bar{v}_{j,k}.$$ Since $$\left(u_k v_k^*\right)_{i,j} = u_{i,k}\bar{v}_{j,k},$$ this is exactly the next line in the proof.