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We have many results on commutativity of two complex matrices. For example, it two matrices are simultaneously diagonalisable then they commute. I would like to know a similar result for super commutativity of matrices.

Let $M_{p|q}(\mathbb{C}) = M_{p|q}(\mathbb{C})_0 \oplus M_{p|q}(\mathbb{C})_1$ be the super algebra of all $(p+q) \times (p+q)$ matrices:

Let $A, B \in M_{p|q}(\mathbb{C})$ (not necessarily homogeneous), If anybody can help me with finding the necessary/sufficient condition for $A$ and $B$ to super commute in terms of eigenvalues or eigenvectors, that would be very helpful to me.

Super commutator is defined as $[X,Y] = XY - (-1)^{|X||Y|}XY$ for all $ X,Y \in M_{p|q}(\mathbb{C})_0 \sqcup M_{p|q}(\mathbb{C})_1$ and extend this definition bilinearly to full $M_{p|q}(\mathbb{C})$. $|X| = \text{homogeneous degree of X}$

Thanks for your thoughts.

Have a good day.

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