When asked to find the Jordan form of an endomorphism I'm usually given a matrix associated with the endomorphism from which I can compute the Jordan, yet this isn't the case with this excercise; instead I'm asked to find the Jordan form of an endomorphism $f$ in $\mathbb{R}^3$ that has the following properties:
- $e_2$ is an eigenvector with eigenvalue $\lambda$.
- $e_3$ is an eigenvector with eigen value $\mu$.
- $x_3=0$ is a $2$-cyclic subspace for some eigenvalue.
- $f(1,1,1)=(2,1,1)$ or, equivalentely, $f(e_1+e_2+e_3)=2e_1+e_2+e_3$.
I'm quite stuck and would appreciate some help.
Based on the given information, the matrix with respect to the canonical basis can be computed from $$ f(e_1)+\lambda e_2+\mu e_3=2e_1+e_2+e_3, \quad f(e_2)=\lambda e_2, \quad f(e_3)=\mu e_3 $$ whence the matrix is \begin{bmatrix} 2 & 0 & 0 \\ 1-\lambda & \lambda & 0 \\ 1-\mu & 0 & \mu \end{bmatrix}
Now you can apply the information you have about the subspace spanned by $e_1$ and $e_2$.