Thread: A strange exercise with matrix norm!

1. A strange exercise with matrix norm!

Let the maximum norm $\displaystyle ||x||_{\infty}=max|x_i|$ (example:$\displaystyle ||(2,-4,1)||_{\infty}=4)$

How can we compute the corresponding matrix norm defined as:

$\displaystyle ||A||_{\infty}=max\frac{\|Ax||_{\infty}}{\|x||_{\i nfty}}$ where x is not equal to 0

if $\displaystyle A=\begin{pmatrix}1 & 2\\3 & -4 \end{pmatrix}$

I would be grateful if someone show me a step-by-step solution.

2. An equivalent way of writing the norm is that $\displaystyle \|A\|_\infty = \max\{\|Ax\|_\infty:\|x\|_\infty\leqslant1\}$. Using that definition, let $\displaystyle x = \begin{bmatrix}a\\b\end{bmatrix}$, with $\displaystyle \max\{|a|,|b|\}\leqslant1$. Then $\displaystyle Ax = \begin{bmatrix}1&2\\3&-4\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}a+2b\\3a-4b\end{bmatrix}$, and $\displaystyle \|Ax\|_\infty = \max\{|a+2b|,|3a-4b|\}$.
Now think about how to maximise that expression subject to the conditions $\displaystyle |a|\leqslant1$, $\displaystyle |b|\leqslant1$, and see if you get the answer 7.