# Left and Right Eigenvectors

• November 27th 2011, 12:58 PM
page929
Left and Right Eigenvectors
Must the left eigenvectors and right eigenvectors of a matrix be the same? Prove or give counter example.

I want to say no, but am not sure if I am even correct let alone proving. Can someone help me out? Thanks!
• November 27th 2011, 03:20 PM
HallsofIvy
Re: Left and Right Eigenvectors
If by "left eigenvalue" you mean a value of $\lambda$ such that
$Av= \begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= \lambda \begin{bmatrix}x \\ y \\ z\end{bmatrix}$
and by "right eigenvalue", you mean a value of $\lambda$ such that
$vA= \begin{bmatrix}x & y & z\end{bmatrix}\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}= \lambda \begin{bmatrix}x & y & z\end{bmatrix}$

Then the "right eigenvalues" of A are the "left eigenvalues" of the transpose of A.