Can diagonal matrix $D \in M_n$ be permuted ($P$) to render a scaled identity matrix, i.e., $\alpha I_n = P D P^*$?

Question

Apologies if it's a trivial question.

Is it possible to permute ($P$) any arbitrary diagonal matrix $D \in M_n$ such that it yields a scaled identity matrix, i.e., $PDP^* = \alpha I_n$, where $\alpha \in F$ is some scalar?

If yes, then can we prove it? If not, then why?

Thank you so much

Answer 1

If I understand your problem correctly, then you are asking if there exists a matrix $P\in\operatorname{Sym}(n)$ s.t. $PDP^T=\alpha I_n,$ where $D$ is a $n\times n$ diagonal matrix. The answer is clearly no, because $PDP^T$ will have the same entries where the rows and columns are shuffled compared to $D$. I'm excluding the fact that all diagonal elements of $D$ are $\alpha,$ of course (trivial).

Answer 2

Only if the diagonal matrix is the scaled identity matrix to begin with. Here's a proof suppose $PDP^*=\alpha I$ where $P$ is Hermitian that is $P^{-1}=P^*$.

Then multiply on the left by $P^*$ to get $DP^*=\alpha P^*$

Then multiply on the right by $P$ to get $D=\alpha I$