I have a $3\times3$ matrix $A$, and compute its SVD $U \Sigma V^\star = A$.
I clamp the singular values in $\Sigma$ to some small range (e.g. $[0.5, 1.5]$ ) and reconstruct matrix $\widetilde{A}=U \widetilde{\Sigma} V^\star$ with clamped singular values. I keep $Det(U), Det(V) \ge 0$, so any mirroring comes across as negative singular values.
For example (in 2D, to simplify) if $A=U \left( \begin{array}{ccc} -1 & 0 \\ 0 & 2 \end{array} \right)V^\star$, then I need to find the matrix $\widetilde{A}=U \left( \begin{array}{ccc} .5 & 0 \\ 0 & 1.5 \end{array} \right) V^\star$. Note that I don't need to know $U$ or $V$. Also, if I clamp approximately, e.g. to $\widetilde{A}=U \left( \begin{array}{ccc} .6 & 0 \\ 0 & 1.4 \end{array} \right) V^\star$, that's fine, too.
Is there some way to do this (perhaps approximately) that is numerically cheaper than computing SVD? Is there a way to do this for $2\times2$ matrix?