1. ## Jordan Normal Forms

Hi, I can answer most questions about the Jordan Normal Form of matrices but there are a couple that I can't think how to approach:

1) (True/False) If nxn matrices A and B (over complex numbers) have the same JNF, so too do A^2 and B^2.

2) (True/False) For any nxn matrices A and B (over complex numbers), AB and BA have the same JNF.

Thanks

2. Originally Posted by Magus01
Hi, I can answer most questions about the Jordan Normal Form of matrices but there are a couple that I can't think how to approach:

1) (True/False) If nxn matrices A and B (over complex numbers) have the same JNF, so too do A^2 and B^2.

Two matrices have the same JNF iff they're similar, so If $\displaystyle J_A,J_B$ are the JNF's of A,B resp., then

$\displaystyle J_A=J_B\Longleftrightarrow A\sim B\Longleftrightarrow A=PBP^{-1}\Longrightarrow A^2=$ $\displaystyle \left(PBP^{-1}\right)^2=PB^2P^{-1}\Longleftrightarrow A^2\sim B^2$

2) (True/False) For any nxn matrices A and B (over complex numbers), AB and BA have the same JNF.

Try to find an (easy!) example s.t. $\displaystyle AB=0$ but $\displaystyle BA\neq 0$ and with different minimal polynomial

Tonio

Thanks
.