A matrix norm...

# Subsection11.1Exercises

##### Exercise11.2

Show that the function that maps $A\in \Rmm$ to $A \rightarrow |\det(A)|$ is not a matrix norm.

##### Exercise11.3

Show that for any of subordinate matrix norm, $\|I\|=1\text{.}$ (That is, the norm of the identity matrix is 1). Is this also true of $\|\cdot\|_F\text{?}$

##### Exercise11.4

Show that for any matrix $A \in \Rmm\text{,}$

1. \begin{equation*} \|A\|_\infty = \max_{i=1, \dots, m} \sum_{j=1}^m |a_{i,j}|. \end{equation*}
2. \begin{equation*} \|A\|_1 = \max_{j=1, \dots, m} \sum_{i=1}^m |a_{i,j}|. \end{equation*}
##### Exercise11.5

Show that for any subordinate matrix norm, $\|A\| \geq |\lambda|$ for any eigenvalue $\lambda$ of $A\text{.}$

##### Exercise11.6

Consider the function $\| \cdot \|:\Rmm \rightarrow \R$ defined by

\begin{equation*} \| A\|_{\widetilde\infty} = \max_{i,j} |a_{ij}|. \end{equation*}

Show that this is a norm. Show that it is not sub-multiplicative.