Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$. Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$. Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.
Let $x$ be a linear recurrent sequence of minimal order $p \geqslant 1$ and minimal polynomial $B$. For any integer $n$ in $\mathbb{N}^*$, we denote by $H_n(x)$ the matrix in $\mathcal{M}_n(\mathbb{K})$ defined by $\forall (i,j) \in \{1,\ldots,n\}^2, [H_n(x)]_{i,j} = x_{i+j-2}$.
Show that the family $(\sigma^k(x))_{0 \leqslant k \leqslant p-1}$ is a basis of $\mathcal{R}_B(\mathbb{K})$.
Deduce from this, for any $n$ in $\mathbb{N}^*$, the rank of the family $(\sigma^k(x))_{0 \leqslant k \leqslant n-1}$.