Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. 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 $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. 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 $x$ is of minimal order $p$ and that the kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.