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I have a question regarding special conditions in matrices. What are all of the special conditions where $AB=BA$ for two square matrices $A$ and $ B$? Would this be asking for multiples of the identity? What are the special conditions? Thank you.

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    $\begingroup$ What do you mean with "all of the special conditions"? Do you look for $\{A\in Mat(n\times n)\mid AB=BA \forall B\in Mat(n\times n)\}$? $\endgroup$ – user302982 Dec 2 '16 at 19:36
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    $\begingroup$ Commuting matrices $\endgroup$ – angryavian Dec 2 '16 at 19:36
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If $B=p(A)$ for some polynomial $p$, then $AB=BA$.

A noteworthy converse case is where $A$ has a Jordan form with only one Jordan block per eigenvalue. In that case, if $AB=BA$ for some $B$, then $B=p(A)$ for some polynomial $p$.

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