TITLE: question on left and right eigenvectors
QUESTION [43 upvotes]: I know that $A\textbf{x}=\lambda \textbf{x}$, where $\textbf{x}$ is right eigenvector, while in $\textbf{y}A =\lambda \textbf{y}$, $\textbf{y}$ is left eigenvector.
But What is significance of left and right eigenvectors ? How do they differ from each other geometrically?

REPLY [14 votes]: The set of left eigenvectors and right eigenvectors together form what is known as a Dual Basis and Basis pair.
http://en.wikipedia.org/wiki/Dual_basis
In simpler terms, if you arrange the right eigenvectors as columns of a matrix B, and arrange the left eigenvectors as rows of a matrix C, then BC = I, in other words B is the inverse of C