Matrix/2x2/Shearing/Characteristic polynomial/Example

We consider the ${}2\times 2$-shearing matrix

${}M={\begin{pmatrix}1&a\\0&1\end{pmatrix}}\,,$

with ${}a\in K$. The characteristic polynomial is

${}\chi _{M}=(X-1)(X-1)\,,$

so that ${}1$ is the only eigenvalue of ${}M$. The corresponding eigenspace is

${}\operatorname {Eig} _{1}{\left(M\right)}=\operatorname {ker} {\left({\begin{pmatrix}0&-a\\0&0\end{pmatrix}}\right)}\,.$

From

${}{\begin{pmatrix}0&-a\\0&0\end{pmatrix}}{\begin{pmatrix}r\\s\end{pmatrix}}={\begin{pmatrix}-as\\0\end{pmatrix}}\,,$

we get that ${}{\begin{pmatrix}1\\0\end{pmatrix}}$ is an eigenvector, and in case ${}a\neq 0$, the eigenspace is one-dimensional (in case ${}a=0$, we have the identity and the eigenspace is two-dimensional). So in case ${}a\neq 0$, the algebraic multiplicity of the eigenvalue ${}1$ equals ${}2$, and the geometric multiplicity equals ${}1$.