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 denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$, and $Q$ is the polynomial of degree $m$ from question 9 such that $Q ( A ) e _ { 0 } = 0$ where $e _ { 0 } = x _ { 0 } - \tilde { x }$. Show that the polynomial $Q$ satisfies $Q ( 0 ) \neq 0$.
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 denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$, and $Q$ is the polynomial of degree $m$ from question 9 such that $Q ( A ) e _ { 0 } = 0$ where $e _ { 0 } = x _ { 0 } - \tilde { x }$.
Show that the polynomial $Q$ satisfies $Q ( 0 ) \neq 0$.