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Assume we have $A, B \in \mathbb{S}_n(\mathbf{R})$, where $\mathbb{S}_n(\mathbf{R})$ is the set of all symmetric matrices with real entries and size $n$. The Lie-bracket (or, the commutator for a matrix group) is defined as follows $$ [A, B] = AB - BA. $$ Now assume $A, B$ are random matrices (feel free to assume any particular ensemble of matrices, say Gaussian ensembles). Are there such "special" random matrices that something could be said about $[A, B]$? Maybe some bounds on its operator norm?

Any other properties along with suggestions and comments will be highly appreciated.


EDIT (Aug 4, 2018): Notice that $(AB)_{ij} = (BA)_{ji}$ yielding the following simple fact: $[A, B]_{ij} = -[A, B]_{ji}$. The latter, in particular, implies that $[A, B]_{ii} = 0$ as noticed by daruma in comments.

So, the lie bracket $[A, B]$ can be rewritten as follows: $$ [A, B] = \begin{pmatrix} 0 & & S \\ & \ddots\\ -S^T & & 0\end{pmatrix} := S_1 - S_2, $$ where $S$ a diagonal block and matrices $S_1$ and $S_2$ can be viewed as upper and lower triangular matrices with corresponding blocks of $S$ and $S^T$, respectively.

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  • $\begingroup$ symmetric matrices are not necessarily commuting. consider $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$. Then $$ [A, B] = AB - BA = \begin{pmatrix} 3 & 6 \\ 1 & 2 \end{pmatrix} - \begin{pmatrix} 3 & 1 \\ 6 & 2 \end{pmatrix} \neq \mathbf{0}_{2 \times 2}. $$ EDIT: @daruma I've corrected my comment, sorry for typos. $\endgroup$ – pointguard0 Aug 1 '18 at 10:41
  • $\begingroup$ Yeah, sorry my mistake. Already deleted the comment before you said it. (But $B$ is not symmetric in that one.) $\endgroup$ – daruma Aug 1 '18 at 10:42
  • $\begingroup$ You can however say that the diagonal elements should be $0$. $\endgroup$ – daruma Aug 1 '18 at 10:42
  • $\begingroup$ $(AB)_{ii}\sum_{j}{a_{ij} b_{ji}}=\sum_{j}{b_{ji}a_{ji}}=(BA)_{ii}$ $\endgroup$ – daruma Aug 1 '18 at 10:44
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    $\begingroup$ Not about symmetric ones but general real valued ones: sciencedirect.com/science/article/pii/S0024379505000832. $\endgroup$ – Rudi_Birnbaum Aug 6 '18 at 10:57

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