Is there a general formula for SVD of inverse matrix?

Question

Suppose $A$ is a square invertible matrix and let $A=USV^{*}$ be a SVD of $A$.

(Note that $U,V$ are unitary and $S$ is diagonal with descending entries on diagonal)

At first, I thought SVD of $A^{-1} = VS^{-1}U^{*}$, but it was wrong since diagonal entries of $S^{-1}$ is not descending order.

I think there is no general formula for SVD of $A^{-1}$, am I right?

Or is there any idea that I don't know?

Thanks in advance!