Please give an example,such that:

$\displaystyle A,B \in M_{n}(\mathbb{F})$

and $\displaystyle AB$ has the same eigenpolynomial as $\displaystyle BA$

but they don't have the same minimal polynomial (Bow)

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- Nov 5th 2009, 02:16 AMXingyuanGive an example
Please give an example,such that:

$\displaystyle A,B \in M_{n}(\mathbb{F})$

and $\displaystyle AB$ has the same eigenpolynomial as $\displaystyle BA$

but they don't have the same minimal polynomial (Bow) - Nov 5th 2009, 03:37 AMHallsofIvy
In order to have the same minimal polynomial, the matrices must have the same eigenvalues. In order not to have the same characteristic polynomial, those eigenvalues must

**not**have the same multiplicity. Try two 3 by 3 matrices with two distinct eigenvalues, one of them of multiplicity 2.

(And you can make this problem really simple by using diagonal matrices.)