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Thread: 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 $\displaystyle l$ is eigenvalue of $\displaystyle A$ it means $\displaystyle A\bold{x} = l\bold{x}$ for some eigenvector $\displaystyle \bold{x}$. Therefore, $\displaystyle A^2\bold{x} = A(A\bold{x}) = A(l\bold{x}) = lA\bold{x} = l^2\bold{x}$. Thus, $\displaystyle \bold{x}$ is an eigenvector for $\displaystyle A$ with eigenvalue $\displaystyle l^2$.
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  3. #3
    MHF Contributor Bruno J.'s Avatar
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    Suppose $\displaystyle \lambda$ is an eigenvalue associated to some vector $\displaystyle v$; then $\displaystyle Av=\lambda v$, and $\displaystyle A^2v=\lambda Av = \lambda^2 v$, so $\displaystyle \lambda^2$ is an eigenvalue of $\displaystyle A^2$.

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