Matrix inversion problem in ridge regression

Question

The ridge estimator can be written in the following way, where the singular value decomposition of X is $X=UDV^{'}$.

I can't quite figure out how the last step (4th step) was obtained from the 3rd one. I think it is something about matrix inverse. Can anyone help me? Thank you.

Answer 1

Since $V$ (and $U$) is a unitary matrix, i.e., $VV^T = I \Leftrightarrow V^{-T} = V$, and $V^TV = I \Leftrightarrow V^T = V^{-1}$, \begin{align} \beta(\lambda) &=\left(VD^2V^T + \lambda V V^T\right)^{-1} VDU^TY \\ &= \left(V\left(D^2 + \lambda I \right)V^T\right)^{-1} VDU^TY \\ &= V^{-T}\left(D^2 + \lambda I \right)^{-1} V^{-1} VDU^TY \\ &= V\left(D^2 + \lambda I \right)^{-1} V^T VDU^TY \\ \end{align}