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Math Help - eigenvector of A+I

  1. #1
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    eigenvector of A+I

    I need to either prove or give a counterexample of the following: If x is an eigenvector of A, then x is also an eigenvector of A+I. I tried several different matrices, and this proposition does seem to be true. However, I have no idea how to prove it.

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  2. #2
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    Suppose x is an eigenvector of A, ie. \exists \lambda \in \mathbb{R} \ : \ Ax = \lambda x \Rightarrow (A + I)x = Ax + x = ...
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  3. #3
    Member kjchauhan's Avatar
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    If x is an eigen vector of a matrix A corresponding to an eigen value \lambda

    \therefore Ax=\lambda x

    \therefore Ax+x=\lambda x+x

    \therefore (A+I)x=(\lambda +1)x

    \therefore (A+I)x=\lambda_1 x, where \lambda_1=\lambda+1

    \therefore x is an eigen vector of a matrix A+I corresponding to an eigen value \lambda_1=\lambda+1.

    Proved.
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