TITLE: Is there a geometrical interpretation to the notion of eigenvector and eigenvalues?
QUESTION [12 upvotes]: The wiki article on eigenvectors offers the following geometrical interpretation:

Each application of the matrix to an arbitrary vector yields a result which will have rotated towards the eigenvector with the largest eigenvalue.

Qn 1: If there is any other geometrical interpretation particularly in the context of a covariance matrix?
The wiki also discusses the difference between left and right eigenvectors.
Qn 2: Do the above geometrical interpretations hold irrespective of whether they are left or right eigenvectors?

REPLY [6 votes]: Instead of giving an answer, let me point out to you this chapter in Cleve Moler's book "Numerical Computing with MATLAB", there is a nice geometric demonstration in MATLAB on how eigenvalues/eigenvectors (as well as singular values/vectors) of an order-2 square matrix are involved in how a circle is transformed into an ellipse after a linear transformation represented by the matrix.