Let $A$, $B$ be $n\times n$ hermitian matrices. Denote by $(\alpha_i)$, $(\beta_i)$, $(\gamma_i)$ the eigenvalues of $A$, $B$, and $A+B$. Assume that there exists permutations $\sigma$, $\tau \in \frak S_n$ so that $$\gamma_i = \alpha_{\sigma(i)} + \beta_{\tau(i)}$$ for all $1\le i \le n$. Does this imply that $A$, $B$ commute?
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No necessarily, Let A and B be two big size nilpotent matrices which are strictly upper triangle – Ali Taghavi Oct 04 '17 at 15:02
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I am sorry I did not pay attention to "hermitian" – Ali Taghavi Oct 04 '17 at 15:04
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may be a related post https://mathoverflow.net/questions/34252/eigenvalues-of-ab-where-a-is-symmetric-positive-definite-and-b-is-diagonal – Ali Taghavi Oct 04 '17 at 15:22
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1@Dennis Serre: I don't know the exact statement of that theorem. Besides, I guess it describes the possibilities for the spectrum of the sum. Not sure it tells when some ( extreme?) cases occur. – orangeskid Oct 04 '17 at 15:45
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this does not look extreme – Fedor Petrov Oct 04 '17 at 16:21
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Please spell my name with one N only: Denis. – Denis Serre Oct 04 '17 at 19:38
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1@DenisSerre You might wish to correct the spelling of Allen Knutson's last name :) – Yemon Choi Oct 04 '17 at 20:31
2 Answers
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The answer is negative. Let $C$ and $D$ be two noncommuting $n\times n$ hermitian matrices, and define the block matrices $$ A=C\oplus (C+D)\oplus 0_n,\ \ \ B=D\oplus(-C)\oplus 0_n, $$ where $0_n$ is the $n\times n$ matrix of all 0's.
Richard Stanley
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This question was the topic of Commutativity and spectra of Hermitian matrices, by Wasin So (1994).
The statement is negative in general: Richard Stanley's counterexample has size $n\geq 6$, a counterexample for $n=3$ is $$A=\begin{pmatrix}0&0&0\\ 0&6-\sqrt 2&2\\ 0&2&4+2\sqrt 2 \end{pmatrix},\;\;B=\begin{pmatrix} 4&0&0\\ 0&-4&0\\ 0&0&0 \end{pmatrix},$$
There are special cases when the statement holds:
If $\gamma_i = \alpha_{\sigma(i)} + \beta_{\tau(i)}$ for a permutation $\sigma(i)$ and $\tau(i)$ of the eigenvalues $\alpha_i,\beta_i,\gamma_i$ in non-increasing order, then the $n\times n$ Hermitian matrices $A,B$ commute if either
- $n=2$
- rank $A=1$
- $\sigma(i)=i$ for all $i$ (identity permutation)
- $\sigma(i)=n+1-i$ for all $i$ (reverse permutation)
Carlo Beenakker
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