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Math Help - Left and Right Eigenvectors

  1. #1
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    Left and Right Eigenvectors

    Must the left eigenvectors and right eigenvectors of a matrix be the same? Prove or give counter example.

    I want to say no, but am not sure if I am even correct let alone proving. Can someone help me out? Thanks!
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  2. #2
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    Re: Left and Right Eigenvectors

    If by "left eigenvalue" you mean a value of [itex]\lambda[/itex] such that
    Av= \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \lambda \begin{bmatrix}x \\ y \\ z\end{bmatrix}
    and by "right eigenvalue", you mean a value of [itex]\lambda[/itex] such that
    vA= \begin{bmatrix}x & y & z\end{bmatrix}\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}= \lambda \begin{bmatrix}x & y & z\end{bmatrix}

    Then the "right eigenvalues" of A are the "left eigenvalues" of the transpose of A.
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