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Suppose we have a matrix $A$ and a symmetric invertible matrix $D$ such that $DA$ is symmetric. The right eigenvectors of $A$ are $v_1,\cdots,v_n$ with eigenvalues $\lambda_1,\cdots, \lambda_n$. Can we use this information to derive (or estimate) left eigenvectors/eigenvalues of $A$?

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I'm going to assume that $D$ is symmetric.

Let $x$ be an eigenvector of $A$ corresponding to eigenvalue $\lambda$. Let $y=Dx$. Then

$$A'y = (A'D)(D^{-1} y) = DAx =\lambda Dx = \lambda y.$$

So then $y$ is an eigenvector of $A'$.

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    $\begingroup$ Thank you; am I right that this proof works even if $D$ is not a diagonal matrix? $\endgroup$ – Hoda Mar 5 '14 at 3:16
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    $\begingroup$ Then I will update the question accordingly. Thanks again! $\endgroup$ – Hoda Mar 5 '14 at 3:21
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    $\begingroup$ Sorry, it has to be symmetric, also! $\endgroup$ – JPi Mar 5 '14 at 3:52
  • $\begingroup$ Oh, right! Thank you, I updated the question again. $\endgroup$ – Hoda Mar 5 '14 at 4:43

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