what is the difference between right and left eigenvector?

Hello,

I read here: https://www.mathworks.com/help/matlab/ref/eig.html that there are two types of eigenvectors, I googled this but with no results. So can any one tell me what is the difference ?

Answer 1

[Stephen23]

http://mathworld.wolfram.com/Eigenvector.html

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Left_and_right_eigenvectors

https://math.stackexchange.com/questions/200370/question-on-left-and-right-eigenvectors

https://math.stackexchange.com/search?q=left+right+eigenvector

etc.

The difference is simply which side they occur on during the multiplication with the matrix A used in their definition.

A right eigenvector as a column vector X_R satisfies:

A * X_R = lambda_R * X_R

A left eigenvector as a row vector X_L satisfies:

X_L * A = lambda_L * X_L

From wikipedia: "Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word 'eigenvector' in the context of matrices almost always refers to a right eigenvector".