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Math Help - Jordan Normal Forms

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by Magus01 View Post
    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 J_A,J_B are the JNF's of A,B resp., then

    J_A=J_B\Longleftrightarrow A\sim B\Longleftrightarrow A=PBP^{-1}\Longrightarrow A^2= \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. AB=0 but BA\neq 0 and with different minimal polynomial

    Tonio

    Thanks
    .
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