I just watched a YouTube video of some professor deriving the pseudo-inverse in terms of the SVD and there is something that I don't understand.
He represents our matrix $A\in \mathbb{R}^{m\times n}$ as $U\Sigma V^T$ where this should be the "economy SVD". Now going from $Ax=b$ he writes the following:
$$V\Sigma^{-1}U^TU\Sigma V^Tx = V\Sigma^{-1}U^Tb$$
I see that $\Sigma^{-1}U^TU\Sigma$ cancels away but what is with $VV^T$?
We are talking about the economy SVD and as far as I understand in general $V$ isn't an orthogonal matrix anymore(in general just a non quadratic matrix with orthonormal columns). Why is $VV^T$ still the identity?