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For a given fixed matrix $M\in SU_{\mathbb{C}}(n)$, how to find all $N\in SO_{\mathbb{C}}(n)$ such that $N^{-1}MN$ is a diagonal matrix?

If we consider a fixed set of $n$ complex vectors $\Gamma:=\{x_1\cdots x_n,x_i\in\mathbb{C}^n\}$, then the above problem is equivalent to finding all matrices $N\in SO_{\mathbb{C}}(n)$ such that it has $\Gamma$ as its eigenvectors(with possible different eigenvalues.) So the totality of such $N$ consists of a subgroup of SO_{\mathbb{C}}(n)$$

One step further, what will the answer change if $N\in O_{\mathbb{C}}(n)$?

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  • $\begingroup$ It's still not quite right, I think. For example, the $N$'s won't normally form a subgroup; $N=1$ typically won't work. $\endgroup$
    – Christian Remling
    Commented Aug 1, 2017 at 22:46
  • $\begingroup$ If $x_j$ are the normalized ev's of $M$, in any order (and written as columns), then you must take $N=(e^{i\alpha_1}x_1,\ldots, e^{i\alpha_n}x_n)$. So the final answer is: those $N$'s of this form that happen to be in $SO(n)$ (if any). If $M$ has multiple eigenvalues, this is still the answer, but you have more choices now. $\endgroup$
    – Christian Remling
    Commented Aug 1, 2017 at 22:50
  • $\begingroup$ @ChristianRemling So there is not a specific way of characterizing all these $N$ besides checking if they actually fall in $SO(n)$? $\endgroup$
    – Henry.L
    Commented Aug 1, 2017 at 22:56
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    $\begingroup$ I'm not sure a highbrow view is helpful here when there's the simple answer "exactly those $N$ that take the standard basis to a basis of ev's of $M$," and then you want only those that are also in $SO(n)$, or at least that's how I would approach it. $\endgroup$
    – Christian Remling
    Commented Aug 1, 2017 at 23:03
  • $\begingroup$ @ChristianRemling Let me sit down and give it a second thought but thanks a lot!! $\endgroup$
    – Henry.L
    Commented Aug 1, 2017 at 23:04

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It is essentially elementary to understand that the centralizer of a diagonal matrix $M$ with entries $m_1,\ldots,m_n$, without loss of generality (conjugating to) have equal entries adjacent to each other, is the group of block-diagonal matrices with block sizes equal to the multiplicities.

Then the matrices conjugating the given diagonal matrix to another diagonal matrix, modulo centralizer, would be permutation matrices.

Combine these...

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