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 $H _ { k + 1 } = H _ { k }$. We denote by $d$ the number of distinct eigenvalues of $A$.
a) In the special case where $r _ { 0 }$ is an eigenvector of $A$, show that the integer $m$ is equal to 1.
b) In the general case, show that $m$ is less than or equal to $d$.
c) For any integer $n$ between 1 and $d$, construct an $x _ { 0 }$ such that the integer $m$ is equal to $n$.
d) Show that the set of $x _ { 0 }$ for which the dimension $m$ is exactly equal to $d$ is the complement of a finite union of sets of the form $\tilde { x } + E$, where $E$ is a vector space of dimension less than or equal to $N - 1$.