If f is an endomorphism of $\mathbb{R}^3$ with only one eigenvalue $\lambda \in \mathbb{R}$ with algebraic multiplicity 3. The matrix of $f$ in standard basis is $A$.
a) What is the characteristic polynomial of $f$?
b) Show all possible forms of Jordan Canonical form for $A$
My answers:
a) Since $\lambda$ is the only eigenvalue and it has algebraic multiplicity = 3, we get:
$$ (n-\lambda)^3 $$
b) \begin{pmatrix} \lambda & 0 & 0\\ 1 & \lambda & 0\\ 0 & 0 & \lambda\\ \end{pmatrix}
\begin{pmatrix} \lambda & 0 & 0\\ 0 & \lambda & 0\\ 0 & 1 & \lambda\\ \end{pmatrix}
\begin{pmatrix} \lambda & 0 & 0\\ 1 & \lambda & 0\\ 0 & 1 & \lambda\\ \end{pmatrix}
Is there anything wrong?! I'm new to this Jordan canonical form...
The three are correct, minus the fact that it is common to put the ones in the upper part. The one you are missing is $$\begin{bmatrix}\lambda &0&0\\0&\lambda&0\\0&0&\lambda \end{bmatrix}. $$