Matrix/2x2/Shearing/Characteristic polynomial/Example

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

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

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

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

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

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

From

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

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