Exercise 2.76

[antodbb]

I'm not sure about the correctness of the formalism, but this is my main idea: extend the smaller bases with null coefficients in the expansion in order to reach a situation similar to that in the standard proof.

[barattolino]

Thanks for this solution.

[antoine-bussy]

Unless I'm mistaken, the proof given in the book already works for the case of different dimensions, if you use the general SVD that also applies to non-square matrices: https://en.wikipedia.org/wiki/Singular_value_decomposition
I don't know why Nielsen and Chuang limited to the square matrices case for SVD... Maybe it is easier to prove?