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 $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ where $\operatorname { deg } ( P ) \in \mathbb { N }$ denotes the degree of the polynomial $P$.
Show that the $H _ { k }$ form a sequence of vector subspaces of $\mathbb { R } ^ { N }$, and show that $H _ { k } \subset H _ { k + 1 }$ for all $k \in \mathbb { N }$.
a) Show that there necessarily exists $k$ such that $H _ { k + 1 } = H _ { k }$. We then denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
b) Show that $\operatorname { dim } \left( H _ { k } \right) = m$ for all $k \geq m$, and that $\operatorname { dim } \left( H _ { k } \right) = k$ for $k \leq m$.