I do not believe the OP is looking for orthonormality of the images of the original basis vectors, only orthogonality, so the result can be (in fact is) true.

This comes down to finding the singular value decomposition. Basically, if

are the singular values of

, then there exist orthonormal bases

and

such that for an arbitrary

, we can write

. One can obtain the singular value decomposition by taking the polar decomposition of

and then applying the finite-dimensional spectral theorem to

.

Yes, I have a feeling the purpose of part 2 was to show the existence of a polar decomposition.