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:

enter image description here

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.

  • $\begingroup$ This seems like pure witchcraft to me, but by golly it works. $\endgroup$ – Alex Jan 27 at 0:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.