Eigenvalues and eigenvectors
Geometric transformations

Eigenvectors and eigenvalues can be useful for understanding linear transformations of geometric shapes.The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.

The characteristic equation for a rotation is a quadratic equation with discriminant D = − 4 (sin θ) 2 {\displaystyle D=-4(\sin \theta)^{2}}, which is a negative number whenever θ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, cos θ ± i sin θ {\displaystyle \cos \theta \pm i\sin \theta }; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.

A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.
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