Let A matrix, vector x, λ eigenvalue
1. Right Eigenvectors and eigenvalues:
A.x = λx
2. Left Eigenvectors and eigenvalues:
x.A = λx
However, be careful that:
If we want to find the left eigenvector associated with the eigenvalue 5, then we find the eigenvector .
This would lead us to see that:
(-1 1 -1).A = (-5 5 -5) = 5. (-1 1 -1)
So, in this example, the eigenvalue 5 has different left and right eigenvectors:
(-1 1 -1) & (1 1 1) respectively.
Remark 1: However, the nice fact about matrices is that always :
left eigenvalue = right eigenvalue.
So we just simply call eigenvalue for short.
《Math Bytes》by Tim Charter
Princeton University Press
[NLB #510 CHA]