Eigenvalue, eigenvector-geometric interpretation in R2
u is an eigenvector of matrix A if its image through A (i.e. A*u), is collinear with u. The corresponding eigenvalue -lambda- is the ratio of the (components of the) vectors Au and u. Because the vectors are collinear, the absolute value of lambda measures the ratio between the lengths of the vectors Au and u.
Move the vector u (drag the point ) till the vectors Au and u becomes collinear. Then u is an eigenvector of matrix A, lambda the corresponding eigenvalue.