Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Points along the horizontal axis do not move at all when this transformation is applied. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation.
Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The linear transformation in this example is called a shear mapping. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The Mona Lisa example pictured here provides a simple illustration. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them.
A 2×2 real and symmetric matrix representing a stretching and shearing of the plane.
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