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Does a diagonal matrix commute with every other matrix of the same size?

I'm stuck on one line of a proof that I am writing, and I would like to switch order between a non-diagonal and a diagonal matrix.

Thanks,

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    $\begingroup$ No. Try with matrices (in lazy notation) $A = (1, 0; 0, 2)$ and $B = (1 ,2; 3 ,4)$. $AB \neq BA$ $\endgroup$
    – Simon S
    Dec 2, 2014 at 4:35
  • $\begingroup$ Ok, got it. Thanks so much, @SimonS. $\endgroup$
    – User001
    Dec 2, 2014 at 4:37
  • $\begingroup$ @SimonS Why not make this an answer to keep this post from getting bumped later? $\endgroup$ Dec 2, 2014 at 5:00

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In general, a diagonal matrix does not commute with another matrix. You can find simple counterexamples in the comments. For a matrix to commute with all the others you need the matrix to be scalar, i.e. diagonal with entries on the diagonal which are all the same.

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  • $\begingroup$ Ok, got it - thanks so much,@Franco. $\endgroup$
    – User001
    Dec 2, 2014 at 7:51

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