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Math Help - Lin Alg Proof

  1. #1
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    Lin Alg Proof

    Given that L is an eigenvalue of an invertible matrix A, prove that 1/L is an eigenvalue for A^(-1). (NOTE: need proof for A in general; giving one example will not suffice)
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  2. #2
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    Quote Originally Posted by caeder012 View Post
    Given that L is an eigenvalue of an invertible matrix A, prove that 1/L is an eigenvalue for A^(-1). (NOTE: need proof for A in general; giving one example will not suffice)
    L is an eigenvalue of A, so there exists an non-zero vector X such that:

    AX=LX

    but A is invertible, so A^{-1} exists, and so:

    A^{-1}[AX]=A^{-1}LX=L A^{-1}X,

    so:

    X= L A^{-1}X,

    or (as L cannot be 0):

    A^{-1}X=(1/L)X.

    Hence (1/L) is an eigenvalue of A^{-1}

    RonL
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