The transformation matrix

$[\begin{array}{cc}2& 1\\ 1& 2\end{array}]$
preserves the direction of vectors parallel to

$[\begin{array}{c}1\\ 1\end{array}]$
(in blue) and

$[\begin{array}{c}1\\ 1\end{array}]$
(in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation.