$T$ is an endomorphism on $ \mathbb{R}^4 $ represented by the matrix $ A $:
$$ A = \begin{pmatrix} -1 & 1 & 0 & 0 \\ -1 & -3 & 0 & 0 \\ 0 & 1 &-1 & 1 \\ 1 & 0 &-1 &-3 \\ \end{pmatrix} $$
The caracteristic polynomial is: $P_A(X) = (X + 2)^4$
The minimal polynomial is: $M_A = (X + 2)^2$
- Give a decomposition of $ \mathbb{R}^4 $ as a direct sum of cyclic subspaces of $T$, with their dimensions.
- Give a Jordan normal form, with its basis.
- Based on the minimal polynomial, the dimension of the cyclic subspaces will be 2, and we will have: $$ \mathbb{R}^4 = <e_1> \oplus <e_2> $$
with: $ <e_{1}> = vect\{ e_{1}, (T + 2I_d)(e_{1}) \} $ and $ <e_{2}> = vect\{ e_{2}, (T + 2I_d)(e_{2}) \} $
- A jordan normal form of $ T $ would be:
$$ J = \begin{pmatrix} -2 & 1 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 &-2 & 1 \\ 0 & 0 & 0 &-2 \\ \end{pmatrix} $$
The basis of $ J $ is the set of vectors:
$ \begin{cases} v_1 = e_1 \\ v_2 = (T + 2I_d)(e_1) \\ v_3 = e_2 \\ v_4 = (T + 2I_d)(e_2) \end{cases} $
Are my answers correct?