Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$. For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1. Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.
Let $A \in M_p(\mathbb{C})$. Let $k \in \mathbb{N}^*$ and $\lambda_1, \lambda_2, \ldots, \lambda_k$ be the roots of $\chi_A$ pairwise distinct, whose respective multiplicities we denote by $n_1, n_2, \ldots, n_k$.
For every integer $q$ between 1 and $p$, we denote by $J_q$ the matrix of $M_q(\mathbb{C})$ whose coefficients are all zero except those located just above the diagonal which equal 1.
Show that, for every $x \in \mathbb{C}$, for every integer $q$ between 1 and $p$, the family $\left\{\left(xI_q + J_q\right)^i,\ 0 \leqslant i \leqslant q-1\right\}$ is free.