I've seen in class and in many different pdf files that the pseudoinverse $-$ i.e for those who don't know functions $f,g$ of finite-dimensional inner product spaces such that $f = fgf$ and $g = gfg$ $-$ are such that $fg$ and $gf$ are by construction self-adjoint.

However the pseudoinverse can also be defined by an SVD, and I would like to used this SVD to show the first definition, in particular to show that $fg$ and $gf$ are indeed self-adjoint.

We can assume that $f = U\sum V^\*$ and $f$'s pseudoinverse is $g = V \sum^+U^\*$. Then I guess we could try to determine $fg$ and $gf$ by calculation, but I've never used SVD before so I'm note quite sure how to properly use $f$ and $g$ to do so. Any idea?

Thanks for your time