# Going from singular value decomposition to Schmidt decomposition

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.