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Suppose $A,B$ are two $n\times n$ complex matrices such that $A,B$ both are singular matrix and also rank $AB$ = rank $BA$. Then is it true that $AB$ and $BA$ will have the same minimal polynomial?

I have tried to find counterexample. But I am not getting.

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1 Answers1

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$A=\begin{pmatrix}0&1&0&1\\0&0&1&0\\0&1&0&0\\0&0&1&0\end{pmatrix},B=\begin{pmatrix}0&-1&1&1\\0&0&0&1\\0&0&0&0\\0&1&0&-1\end{pmatrix}$.

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    It is natural to set up the equations of $AB=J_2(0)\oplus J_2(0)$ and $BA=J_3(0)\oplus0$, but how did you solve them? – user1551 Jun 29 '19 at 10:25
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    @user1551 , I ask Maple for the Grobner's basis of the system $A=[a_{i,j}],B=[b_{i,j}],AB=diag(J_2,J_2),BA=diag(J_3,J_1)$; there are $7$ free parameters but we do not know what they are; we use a trial and error approach.

    ;

     –  Jun 29 '19 at 10:38
  • I see. Thanks for the explanation. – user1551 Jun 29 '19 at 10:45