How to find an unitary transformation of $A$ that minimize $(A'_{i,i}-1)^2$?

Question

Is there a way to find an unitary transformation $$ A'=U^+AU $$ that minimize: $$(A'_{i,i}-1)^2$$

In other words, the diagonal elements must be similar to one: $A'_{i,i} \approx 1$

Any hint?

Thank you!

Answer 1

I will assume in the following that $A$ is a hermitian matrix. Then the matrix $A' = U ^{\dagger} A U$ is again hermitian.

Let set of possible diagonals value $ (d_1, \ldots d_n)$ achieved by the matrices of form $A' = ^{\dagger} A U$ forms a polytope in the hyperplane $\sum d_i = \text{trace} A$. This is an important theorem of Horn.

The closest point to the point $(1,\ldots, 1)$ is the one with all coordinates equal. It can be obtained for $U = \frac{1}{\sqrt{n}} (\exp 2 \pi i\cdot k\cdot l /n)_{0\le k,l \le n-1}$