# Left Eigenvectors vs. Right Eigenvectors

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 $A$?

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 $A'$.
• Thank you; am I right that this proof works even if $D$ is not a diagonal matrix? – Hoda Mar 5 '14 at 3:16