Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$. Show that there exists a polynomial $Q$ of degree $m$ such that $Q ( A ) e _ { 0 } = 0$, where $e _ { 0 } = x _ { 0 } - \tilde { x }$.
Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$,
$$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$
We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
Show that there exists a polynomial $Q$ of degree $m$ such that $Q ( A ) e _ { 0 } = 0$, where $e _ { 0 } = x _ { 0 } - \tilde { x }$.