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Math Help - A strange exercise with matrix norm!

  1. #1
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    A strange exercise with matrix norm!

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

    How can we compute the corresponding matrix norm defined as:

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

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

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

    Thanks in Advance
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  2. #2
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    An equivalent way of writing the norm is that \|A\|_\infty = \max\{\|Ax\|_\infty:\|x\|_\infty\leqslant1\}. Using that definition, let x = \begin{bmatrix}a\\b\end{bmatrix}, with \max\{|a|,|b|\}\leqslant1. Then Ax = \begin{bmatrix}1&2\\3&-4\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}a+2b\\3a-4b\end{bmatrix}, and \|Ax\|_\infty = \max\{|a+2b|,|3a-4b|\}.

    Now think about how to maximise that expression subject to the conditions |a|\leqslant1, |b|\leqslant1, and see if you get the answer 7.
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