Show all $n\times n$ matrices which $AB=BA$ have a common eigenvector

Asked Apr 25 '21 at 18:18

Active Apr 25 '21 at 18:18

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If $S$ is the set of all complex $n \times n$ matrices and for any two arbitrary $A$ and $B$ in $S$ we have $AB=BA$ then all matrices in $S$ have a common eigenvector.

I can show the result for two digonalizable matrix. But I don't know how to show that for all matrices in $S$. Also there is no condition of being digonalizable in this question.

asked Apr 25 '21 at 18:18

User558256555

@DietrichBurde The matrices here aren't assumed to be diagonalisable, so I don't think that that answer applies. – Aryaman Maithani Apr 25 '21 at 18:28
1

@AryamanMaithani See this duplicate (same proof as in the other duplicate, but they really call the common eigenvector $y$). Do you agree? – Dietrich Burde Apr 25 '21 at 18:45
@DietrichBurde: Yes, that seems more appropriate. – Aryaman Maithani Apr 25 '21 at 19:20

Show all $n\times n$ matrices which $AB=BA$ have a common eigenvector

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