Is it possible to obtain right eigenvectors from left eigenvectors under certain conditions?

Question

Suppose we solved the eigenvalue problem $VA=\Lambda V$ and the resulting matrix of left eigenvectors $V$ is invertible. Then, diagonalize $A=V^{-1}\Lambda V$, multiply both sides by $V^{-1}$ to get $AV^{-1}=V^{-1}\Lambda$. Thus, the inverse of $V^{-1}$ is the matrix of right eigenvectors. I would like to confirm that my reasoning is correct.

Answer 1

Your reasoning is correct, $V^{-1}$ contains the right eigenvectors.

In general, left and right eigenvectors need not to be orthogonal: Take $A=I_2$, then $$ \begin{pmatrix}1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 1\end{pmatrix} $$ are left eigenvectors, $$ \begin{pmatrix}1 \\ 1\end{pmatrix}, \begin{pmatrix}1 \\ 2\end{pmatrix} $$ are right eigenvectors, but there is no orthogonality.