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Math Help - eigenvalues of qudratic matrix

  1. #1
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    eigenvalues of qudratic matrix

    Let l
    1, l2 . . . , ln be the eigenvalues of a matrix A. What are the eigenvalues of A^2?

    there must be some relation bwn. the eigenvalues of A and A^2 matrix which i cannot see it.. Please assist.

    Many thanks.

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  2. #2
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    Quote Originally Posted by saskadimova View Post
    Let l
    1, l2 . . . , ln be the eigenvalues of a matrix A. What are the eigenvalues of A^2?

    there must be some relation bwn. the eigenvalues of A and A^2 matrix which i cannot see it.. Please assist.

    Many thanks.

    If l is eigenvalue of A it means A\bold{x} = l\bold{x} for some eigenvector \bold{x}. Therefore, A^2\bold{x} = A(A\bold{x}) = A(l\bold{x}) = lA\bold{x} = l^2\bold{x}. Thus, \bold{x} is an eigenvector for A with eigenvalue l^2.
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    MHF Contributor Bruno J.'s Avatar
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    Suppose \lambda is an eigenvalue associated to some vector v; then Av=\lambda v, and A^2v=\lambda Av = \lambda^2 v, so \lambda^2 is an eigenvalue of A^2.

    Note that the converse does not always hold; A could have no eigenvalues while A^2 could. For instance if A rotates the plane by an angle of \pi/2, then A has no (real) eigenvalues but A^2 does.
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