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Eigentheory/R^2/Elementary/Example

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A linear mapping from to is described by a -matrix with respect to the standard basis. We consider the eigenvalues for some elementary examples. A homothety is given as , with a scaling factor . Every vector is an eigenvector for the eigenvalue , and the eigenspace for this eigenvalue is the whole . Beside , there are no other eigenvalues, and all eigenspaces for are . The identity only has the eigenvalue .

The reflection at the -axis is described by the matrix . The eigenspace for the eigenvalue is the -axis, the eigenspace for the eigenvalue is the -axis. A vector with is not an eigenvector, since the equation

does not have a solution.

A plane rotation is described by a rotation matrix for the rotation angle , For , this is the identity, for , this is a half rotation, which is the reflection at the origin or the homothety with factor . For all other rotation angles, there is no line sent to itself, so that these rotations have no eigenvalue and no eigenvector (and all eigenspaces are ).