(Solved): Show that given any consistent matrix norm \( \|\cdot\| \), then \( \rho(\boldsymbol{A}) \leq\|\bol ...

Show that given any consistent matrix norm \( \|\cdot\| \), then \( \rho(\boldsymbol{A}) \leq\|\boldsymbol{A}\| \) for all \( \boldsymbol{A} \in \mathbb{R}^{n \times n} \). The following property holds for consistent matrix norms: Given any matrix \( A \in \mathbb{R}^{n \times n} \) and \( \epsilon>0 \), then there exists a consistent matrix norm \( \|\cdot\|_{\epsilon} \) such that, \[ \|\boldsymbol{A}\|_{\epsilon} \leq \rho(\boldsymbol{A})+\epsilon \] Combining your proof along with this property we can state that, \[ \rho(\boldsymbol{A})=\min _{\|\cdot\|}\|\boldsymbol{A}\| \] Using (11) show that if \( \boldsymbol{A} \) is a square matrix then, \[ \lim _{k \rightarrow \infty} \boldsymbol{A}^{k}=0 \Longleftrightarrow \rho(\boldsymbol{A})<1 \] d) [Bonus] Show that: \[ \frac{1}{1+\|\boldsymbol{A}\|} \leq\left\|(\boldsymbol{I}-\boldsymbol{A})^{-1}\right\| \leq \frac{1}{1-\|\boldsymbol{A}\|} \] where \( \|\cdot\| \) is an induced matrix norm such that \( \|A\|<1 \) (Note: The results of this proof are useful in deriving the bounds on the change in solution when the ystem \( \boldsymbol{A x}=\boldsymbol{b} \) is perturbed to \( (\boldsymbol{A}+\delta \boldsymbol{A}) \boldsymbol{x}=\boldsymbol{b}+\delta \boldsymbol{b} \). The corresponding bound was discussed in class and ou can find it in the lecture notes.) \( \rho(\boldsymbol{A})=\min _{\|\cdot\|}\|\boldsymbol{A}\| \)