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    eigenvalues

    Show that A and A^{T} have the same eigenvalues. what if anything can we say about the associated eigenvectors of A and A^{T}?
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    Quote Originally Posted by antman View Post
    Show that A and A^{T} have the same eigenvalues. what if anything can we say about the associated eigenvectors of A and A^{T}?
     \det (A - \lambda I) = \det (A - \lambda I)^T = \det (A^T - \lambda I) .


    The eigenvectors of A^T are generally different to those of A. However, there is a relationship between the two:

    If v_i is an eigenvector of A corresponding to the eigenvalue \lambda_i and w_j is an eigenvector of A^T corresponding to the eigenvalue \lambda_j then v_i^T w_j = 0 ~ (\lambda_i \neq \lambda_j).

    Proof:

    A v_i = \lambda_i v_i \Rightarrow  v_i^T A = \lambda_i v_i^T \Rightarrow v_i^T A w_j = \lambda_i v_i^T w_j .... (1)

    A^T w_j = \lambda_j w_j \Rightarrow v_i^T A^T w_j = \lambda_j v_i^T w_j .... (2)

    (1) - (2): 0 = \lambda_i v_i^T w_j - \lambda_j v_i^T w_j and the result is easily seen.
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