Left/right eigenvectors

Quote:

Originally Posted by rak

A fairly simple one (but one I can't seem to find a definition for anywhere), so sorry if it shouldn't be in the advanced section.

What is the difference between right and left eigenvectors?

For example I have

(2/3 1/3)
(1/6 5/6)

Found eigenvalues 1/2 and 1, and eigenvectors (-2,1) (1,1) respectively. But which is the right eigenvector?!

Thanks

what you've found are right eigenvectors because you solved $A \bold{x}= \lambda \bold{x},$ where $A$ is your matrix, $\lambda= 1, \frac{1}{2},$ and $\bold{x}$ is a column. to find left eigenvectors you need to solve $\bold{x}A=\lambda \bold{x},$ for the same

matrix and eigenvalues but this time your eigenvector, $\bold{x},$ is a row. you'll get $[1 \ \ 2]$ and $[1 \ \ -1].$ so basically the left eigenvectors of $A$ are the transpose of the right eigenvectors of $A^T.$