How to see this matrix equality When doing some formulas for regression I encountered this which I think is true by trying some examples:
$$X^T (X X^T + \lambda I)^{- 1} = (X^T X + \lambda I)^{- 1} X^T$$
Here $\lambda > 0$. I'm stuck on how to prove this. Help/counterexamples appreciated.

By multiplying from the left with $(X^T X + \lambda I)$ and from the right with $(X X^T + \lambda I)$, we obtain
$$
  (X^TX + \lambda I) X^T = X^T (X X^T + \lambda I),
$$
and from here it's obvious.
