Provided two diagonal real matrix which has positive entries, $V$ and $U$.

Find a real matrix $A$, satisfying $A^TA=a^2I$ for some scalar $a$, to minimise

$\left|A^TVA-U\right|\quad\quad(*)$

where the matrix norm could be an induced one, or in form of $|M|^2_{F}=\mathrm{tr}(M^TM)$.

I believe the problem is quite useful, however I am not sure where I can find the related materials. A numerical approach is also welcome.

I found some related works , I think I can program the general framework for non-linear optimisation problem with unitary constraints. But since $(*)$ is only a quadratic form. I wonder if there are some improvements.

remark: there are two trivial cases, namely $V=U$ or $U=I$.