Definition11.1
A matrix norm...
A matrix norm...
Show that the function that maps \(A\in \Rmm\) to \(A \rightarrow |\det(A)|\) is not a matrix norm.
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 \(\|\cdot\|_F\text{?}\)
Show that for any matrix \(A \in \Rmm\text{,}\)
Show that for any subordinate matrix norm, \(\|A\| \geq |\lambda|\) for any eigenvalue \(\lambda\) of \(A\text{.}\)
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.