Eigenvectors & Eigenvalues

Eigenvectors are the special directions A only stretches — A·v = λv. They are drawn in blue, scaled in orange.

A
×
Eigenvalues
λ1 = 1λ2 = 3
Eigenvectors
[0.70710.7071−0.70710.7071]\begin{bmatrix} 0.7071 & 0.7071 \\ -0.7071 & 0.7071 \end{bmatrix}
Reading the picture

Faint gray = input vectors on a circle. Faint orange = A·v. Blue lines mark eigenvector directions — input and output are colinear there.