We consider the
-shearing matrix
-
![{\displaystyle {}M={\begin{pmatrix}1&a\\0&1\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f800f1618170ca30e208121aab40ef9152c3327b)
with
.
The
characteristic polynomial
is
-
![{\displaystyle {}\chi _{M}=(X-1)(X-1)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/598ddb0bb210cefc1a331ecfc36a8337830c9083)
so that
is the only
eigenvalue
of
. The corresponding
eigenspace
is
-
![{\displaystyle {}\operatorname {Eig} _{1}{\left(M\right)}=\operatorname {ker} {\left({\begin{pmatrix}0&-a\\0&0\end{pmatrix}}\right)}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca7eb6a6fb7fd6d1bd0b2416cb8144f986e7d4bc)
From
-
![{\displaystyle {}{\begin{pmatrix}0&-a\\0&0\end{pmatrix}}{\begin{pmatrix}r\\s\end{pmatrix}}={\begin{pmatrix}-as\\0\end{pmatrix}}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd1a0ce10eae380cd56b0b22857d2666b9b9365)
we get that
is an
eigenvector,
and in case
,
the eigenspace is one-dimensional
(in case
,
we have the identity and the eigenspace is two-dimensional).
So in case
,
the
algebraic multiplicity
of the eigenvalue
equals
, and the
geometric multiplicity
equals
.