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Math Help - Similar Matrices

  1. #1
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    Similar Matrices

    Little bit stuck on this problem.

    Let A and B be similar Matrices. Let \lambda be an eigenvalue of A and B
    Show that dim Ker(A-\lambda I) = dim Ker(B-\lambda I)

    Any help would be great. Thanks!
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  2. #2
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    Quote Originally Posted by joe909 View Post
    Little bit stuck on this problem.

    Let A and B be similar Matrices. Let \lambda be an eigenvalue of A and B
    Show that dim Ker(A-\lambda I) = dim Ker(B-\lambda I)

    Any help would be great. Thanks!

    First show that if P is an invertible matrix, then \dim A=\dim PA=\dim AP , for any other matrix A .

    Next, we have that A=P^{-1}BP , for some invertible matrix P (why?) , so by the above :

    \dim\ker(A-\lambda I)=\dim\ker P(A-\lambda I)P^{-1}=\dim\ker(B-\lambda I) . Justify and explain each step.

    Tonio
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