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 define $x _ { 0 } + H _ { k }$ as the subset of points of $\mathbb { R } ^ { N }$ of the form $x _ { 0 } + x$ where $x$ ranges over the vector space $H _ { k }$. a) Show that $\tilde { x } \in x _ { 0 } + H _ { m }$. b) Show that, for all $k \in \{ 0 , \ldots , m - 1 \}$, we have $\tilde { x } \notin x _ { 0 } + H _ { k }$.
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 define $x _ { 0 } + H _ { k }$ as the subset of points of $\mathbb { R } ^ { N }$ of the form $x _ { 0 } + x$ where $x$ ranges over the vector space $H _ { k }$.
a) Show that $\tilde { x } \in x _ { 0 } + H _ { m }$.
b) Show that, for all $k \in \{ 0 , \ldots , m - 1 \}$, we have $\tilde { x } \notin x _ { 0 } + H _ { k }$.