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?