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Math Help - Similar matrices

  1. #1
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    Similar matrices

    Hi

    Are these 2 matrices similar to each other?

    A= [0 5 0]
    .....[0 0 5]
    .....[0 0 0]

    B = [0 10 0 ]
    ......[0 0 10]
    ......[0 0 0]

    They have the same eigenvalues of 0,0,0 and the same rank, but there is no P or P^-1 that you can find to give you a diagonal matrix, being that the P you would get is entirely singular. Does that entail then that A and B here aren't similar at all? Thanks
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  2. #2
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    Quote Originally Posted by lute View Post
    Hi

    Are these 2 matrices similar to each other?

    A= [0 5 0]
    .....[0 0 5]
    .....[0 0 0]

    B = [0 10 0 ]
    ......[0 0 10]
    ......[0 0 0]

    They have the same eigenvalues of 0,0,0 and the same rank, but there is no P or P^-1 that you can find to give you a diagonal matrix, being that the P you would get is entirely singular. Does that entail then that A and B here aren't similar at all? Thanks
    Try taking P to be a diagonal matrix, say P = \begin{bmatrix}x&0&0\\ 0&y&0\\ 0&0&z\end{bmatrix}. Calculate the product P^{-1}AP and see if you can make it equal to B, for suitable values of x,y and z.
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  3. #3
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    Thanks! I was indeed able to find a P and a P^-1 that was able to get A equal to B, which means that they are similar matrices afterall.
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