[Math] Left Eigenvectors vs. Right Eigenvectors

eigenvalues-eigenvectorslinear algebra

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 $A$?

Best Answer

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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