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Section11Matrix Norms (NM, 26 Sep)

Definition11.1

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.