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Math Help - Eigenvalues/Diagonalization

  1. #1
    fkf
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    Eigenvalues/Diagonalization

    A is a 5x5 matrix with two eigenvalues. One eigenspace is three-dimensional, and the other one is two-dimensional. Is A diagonalizable? Why?
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    Re: Eigenvalues/Diagonalization

    yes. the sum total of the dimensions of the eigenspaces equals the dimension of the domain of A. therefore, we have an eigenbasis for the domain of A. if a basis for the eigenspace corresponding to λ1 is {v1, v2, v3} and a basis for the eigenspace corresponding to λ2 is {v4, v5}, then the matrix P whose columns are the eigenbasis for V satisfies:

    AP = PD, where D = diag{λ1, λ1, λ1, λ2, λ2} so that:

    P-1AP = D, a diagonal matrix.

    the invertibility of P is guaranteed by the linear independence of the eigenvectors (the only vector Eλ1 and Eλ2 have in common is the zero-vector:

    if Av = λ1v = λ2v, then:

    1 - λ2)v = 0, since λ1 ≠ λ2, we must have v = 0).
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