# Left/right eigenvectors

• May 18th 2009, 03:54 AM
rak
Left/right eigenvectors
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
• May 18th 2009, 04:22 AM
NonCommAlg
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 $\displaystyle A \bold{x}= \lambda \bold{x},$ where $\displaystyle A$ is your matrix, $\displaystyle \lambda= 1, \frac{1}{2},$ and $\displaystyle \bold{x}$ is a column. to find left eigenvectors you need to solve $\displaystyle \bold{x}A=\lambda \bold{x},$ for the same

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