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Math Help - Question about eigenvectors

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    Question about eigenvectors

    If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?
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  2. #2
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    Quote Originally Posted by nicolem1051 View Post
    If v is an eigenvector of A with corresponding eigenvalue λ, show that v is an eigenvectors of A^-1. What is its corresponding eigenvalue?
    This should be rather easy. The notation, however, depends on whether you write vectors in rows or in columns.
    If I use row vectors, then v being eigenvector means
    v.A=\lambda.v.
    What do you get if you multiply this equality by A^{-1} from the right?
    (Obviously, you assume that A is invertible, even if you did not write this assumption explicitly.)
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  3. #3
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    A\mathbf{x}=\lambda\mathbf{x}

    Multiple by A inverse on the left and move lambda left since it is a scalar.

    A^{-1}A\mathbf{x}=\lambda A^{-1}\mathbf{x}

    A inverse A is I.

    \mathbf{x}=\lambda A^{-1}\mathbf{x}

    Divide by lambda.

    \frac{1}{\lambda}\mathbf{x}=A^{-1}\mathbf{x}
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