Eigentheory/R^2/Elementary/Example

A linear mapping from ${}\mathbb {R} ^{2}$ to ${}\mathbb {R} ^{2}$ is described by a ${}2\times 2$-matrix with respect to the standard basis. We consider the eigenvalues for some elementary examples. A homothety is given as ${}v\mapsto av$, with a scaling factor ${}a\in \mathbb {R}$. Every vector ${}v\neq 0$ is an eigenvector for the eigenvalue ${}a$, and the eigenspace for this eigenvalue is the whole ${}\mathbb {R} ^{2}$. Beside ${}a$, there are no other eigenvalues, and all eigenspaces for ${}\lambda \neq a$ are ${}0$. The identity only has the eigenvalue ${}1$.

The reflection at the ${}x$-axis is described by the matrix ${}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$. The eigenspace for the eigenvalue ${}1$ is the ${}x$-axis, the eigenspace for the eigenvalue ${}-1$ is the ${}y$-axis. A vector ${}(s,t)$ with ${}s,t\neq 0$ is not an eigenvector, since the equation

${}(s,-t)=\lambda (s,t)\,$

does not have a solution.

A plane rotation is described by a rotation matrix ${}{\begin{pmatrix}\operatorname {cos} \,\alpha &-\operatorname {sin} \,\alpha \\\operatorname {sin} \,\alpha &\operatorname {cos} \,\alpha \end{pmatrix}}$ for the rotation angle ${}\alpha$, ${}0\leq \alpha <2\pi$ For ${}\alpha =0$, this is the identity, for ${}\alpha =\pi$, this is a half rotation, which is the reflection at the origin or the homothety with factor ${}-1$. 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 ${}0$).