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Let $A$ and $B$ be $n\times n$ matrices over $\mathbb{C}$. If $AB=BA$, we know that we can simultaneously diagonalize $A$ and $B$ (or make them Jordan canonical form). What if they are weakly commutative in the sense that $AB=cBA$ for some $c\in \mathbb{C}^{\times}$? What can we say about $A$ and $B$?

I am sorry being a bit vague, but I came across with this kind of matrices in my little project and wonder what we can say about them.

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    Have you tried looking where the proofs of the properties that are true when $AB = BA$ get stuck (if they do) ?2012-11-24
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    @Sheng Please remove the abstract-algebra tag which doesn't fit here.2012-11-25
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    May I ask what is this $\mathbb{C}^{\times}$ thing?2012-11-25
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    As Martin says below $AB=cBA$ for $c\ne1$ imposes strong constraints. For example, you can take determinant, traces etc of the noncommutativity equation.2012-11-25

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