We keep the notations from Parts II and III. We denote $e _ { k } = x _ { k } - \tilde { x }$ and $e _ { 0 } = x _ { 0 } - \tilde { x }$. We recall that $I _ { N }$ is the identity matrix of order $N$, and $\| \cdot \|$ denotes the matrix norm defined in question 2. Show that $$\left\| e _ { k } \right\| _ { A } \leq \left\| e _ { 0 } \right\| _ { A } \min \left\{ \left\| I _ { N } + A Q ( A ) \right\| \mid Q \in \mathbb { R } [ X ] , \operatorname { deg } ( Q ) \leq k - 1 \right\}$$ (One may use the properties of $A ^ { 1 / 2 }$ demonstrated in question 6.)
We keep the notations from Parts II and III. We denote $e _ { k } = x _ { k } - \tilde { x }$ and $e _ { 0 } = x _ { 0 } - \tilde { x }$. We recall that $I _ { N }$ is the identity matrix of order $N$, and $\| \cdot \|$ denotes the matrix norm defined in question 2.
Show that
$$\left\| e _ { k } \right\| _ { A } \leq \left\| e _ { 0 } \right\| _ { A } \min \left\{ \left\| I _ { N } + A Q ( A ) \right\| \mid Q \in \mathbb { R } [ X ] , \operatorname { deg } ( Q ) \leq k - 1 \right\}$$
(One may use the properties of $A ^ { 1 / 2 }$ demonstrated in question 6.)