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Math Help - Similar Matrices

  1. #1
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    Similar Matrices

    1.) Given two matrices, A and B, show that
    A = [[1,1],[-1,4]], and
    B =[[2,1],[1,3]] are similar.

    Then, show that the matrix
    C = [[3,1],[-6,-2]] and
    D = [[-1,2],[1,0]] are NOT similar.

    Could you check to see if I am right? The theorem for similar matrices states that if n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

    Or: Two n n matrices A and B are similar, if there exists a non-singular n n matrix P such that B = P−1AP.

    For A,

    (1*4) - (-1*1) = 5, so it is non-singular.

    And B,

    (2*3) - (1*1) = 5, so it is non-singular

    (A - LI)x = 0

    A = [[1-L, 1],[-1,4-L]]

    (1-L)*(4-L) - (-1) = 0

    Solving for L, we get the eigenvalues are 5/2+1/2*sqrt(5), 5/2-1/2*sqrt(5)

    B = [[2-L, 1],[1, 3-L]]

    (2-L)*(3-L) - 1*1 = 0

    Solving for L, we get the eigenvalues are the same as above.

    Is this sufficient to show that they are similar, since they both have the same eigenvalues?

    To show C and D are not similar:

    For C:

    (3-L)*(-2-L) - (-6) = 0

    Solving for L, we get the eigenvalues are 0 and 1.

    For D:

    (-1-L)*(0-L) - 2 = 0

    Solving for L, we get the eigenvalues are 1 and -2.

    Is this sufficient to prove that they are not similar, since they don't have the same eigenvalues?
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  2. #2
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    Apparently its sufficient enough to show that the eigenvalues are not the same to show the matrices are not similar; however, it is not sufficient enough to show that if the eigenvalues are the same, the matrices are similar. How would I do it with the P^(-1)AP thing?
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