I was reading this book on numeric linear algebra and it said pseudo inverse of a singular value decomposition (SVD) is equal to it's "real" inverse for a square matrix. It said it is quite clear that they are equal but I don't really understand how. I know pseudo inverse of a invertible matrix $A* = V\Sigma^{-1} U^T$. But how this is equal to inverse of matrix $A$.
The SVD of a Matrix $A = U\Sigma V^T$, so it's inverse $A^{-1} = (U\Sigma V^T)^{-1}$, how does this equal $V\Sigma^{-1}U^T$?
Maybe I am missing some basic inverse calculation? Can somebody please show this to me?
P.S: This is my first question here, sorry if I made any mistakes