Consider the transformation that takes (x, y) to (2x+y, x+2y). (1, 0) goes to (2, 1) which is a turn. Likewise (0, 1) goes to (1, 2). (1, 1), however goes to (3, 3) which is in the same direction and is thus not turned, but merely lengthened by three. (1, 1) is an eigenvector and its eigenvalue is 3. (1, −1) goes to (1, −1) thus (1, −1) is an eigenvector with eigenvalue 1. Note that the two eigenvectors are orthogonal. If we choose them as basis vectors then the transformation’s matrix is:

3	0
0	1

instead of

2	1
1	2