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I wasn't $100\%$ sure in the accuracy of what I obtained, so may I ask for verification?


Let $A,B\in M_n(\Bbb C)$ be two similar Hermitian matrices. Then $A=P^{-1}BP$.

According to the spectral theorem, every Hermitian matrix is diagonalizable, so $A=U_1^{-1}DU_1$ and $B=U_2^{-1}DU_2$, where $U_1,U_2\in M_n(\Bbb C)$ are unitary and $D=\left(\delta_{ij}\right)\in M_n(\Bbb C)$ is diagonal s. t. $\delta_{ii}\in\sigma(A)=\sigma(B)$.

Since $A$ and $B$ are unitarily similar to $D$, I wanted to write $A$ in the following form: $$A=P^{-1}U_2^{-1}DU_2P$$ Then $U_2P=U_1\implies P=U_2^{-1}U_1$. Since both $U_1$ and $U_2^{-1}$ are unitary, $P$, as a product of two unitary matrices, is also unitary.

Question according to this result:

Are all similar Hermitian matrices unitarily similar?

If this is valid, can it be used in the proof that the matrix representation of a Hermitian operator in an orthonormal basis is a Hermitian matrix? The statement is obvious when, given an arbitrary Hermitian matrix, one is asked to diagonalize it. I thought $P$ could be the transition matrix from one orthonormal basis to another one.


Thank you in advance!

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  • $\begingroup$ I am having a hard time following what you are saying. It is true that every Hermitian matrix is unitarily equivalent to a diagonal matrix. But two diagonal matrices are unitarily equivalent if and only if they have the same diagonal elements, up to writing them in a different order. So the diagonal matrix whose diagonal entries are 1, 1, and 2 is not unitarily similar to the diagonal matrix whose diagonal entries are 2, 2, and 1. $\endgroup$ Aug 17 '20 at 20:22
  • $\begingroup$ @StephenMontgomery-Smith, I didn't mean unitarily equivalent, but unitarily similar. I'm wondering if Hermitian matrices are similar, are they unitarily similar? Let's say we take your example of a diagonal matrix with entries $\operatorname{diag}\{1,1,2\}$. Then, I mean, $A$ is unitarily similar to $D$ and $B$ is unitarily similar to $D$. Then, my problem boils to the possible transitivity of the relation "being unitarily similar". I hope it sounds less messy. We used the term equivalent only for the matrices of the same type and rank to distinguish the similarity. $\endgroup$
    – Invisible
    Aug 17 '20 at 20:31
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    $\begingroup$ Sorry. I thought that similar and equivalent meant the same thing in this context. $\endgroup$ Aug 17 '20 at 20:34
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    $\begingroup$ Anyway, is your question this? "If two Hermitian matrices are similar, then they are unitarily similar?" If that is the question, then the answer is "yes." $\endgroup$ Aug 17 '20 at 20:34
  • $\begingroup$ @StephenMontgomery-Smith, thank you for verifying! And more importantly, for the patience! $\endgroup$
    – Invisible
    Aug 17 '20 at 20:35
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As verifed by @StephenMontgomery-Smith, all similar Hermitian matrices are unitarily similar, indeed.

Howerver, $P$ doesn't necessarily have to be unitary if, e. g., $A=B=0$. Then every invertible $P$ would suffice.


You may see this parallel question I posted on Quora, answered yesterday by Aaron Dunbrack.

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