How to derive the following matrix equation?

Question

The SVD of a symmetric matrix $X$ is as follows: $$ [U, \Lambda, U^T] = svd(X); $$ then, if we have the following matrix: $$ Y = {\left(U(\Lambda+\alpha I)U^T\right)}^{-1} U \Lambda $$ My question is: how to derive the following equation? $$ Y = U \frac{\Lambda}{\Lambda+\alpha I} $$

Could you help give some advice, please?

Answer 1

So we have $Y = (U(\Lambda + \alpha I)U^T)^{-1}U\Lambda$. Invert the left hand side, giving us:

$$Y = ((U^T)^{-1}(\Lambda+\alpha I)^{-1}U^{-1})U\Lambda = (U(\Lambda + \alpha I)^{-1}U^{-1})U\Lambda$$ which you should be able to cancel and get your result.

Answer 2

Since $X$ is symmetric, $U$ is orthogonal. Just apply $(AB)^{-1} = B^{-1}A^{-1}$ and $U^T = U^{-1}$.