The $3$-norm on $\mathbb R^n$ is defined as
$$\|x\|_3 := \sqrt[3]{|x_1|^3+\dots+|x_n|^3}$$
The natural matrix norm it induces on $\mathbb R^{n \times n}$ is
$$\|A\|_3 = \max_{\|x\|_3=1} \|Ax\|_3$$
For $y \in \mathbb R$, let
$$A_y = \left(\begin{matrix} 1 & y \\ 0 & 1 \end{matrix}\right)$$
Give a table showing $\|A_y\|_3$ for $y = 1, \dots, 9$.
Matrix $3$-norm
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matrices
optimization
norm
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0Since it's about numerical method, you have to write a program. What have you done? – 2012-04-28
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0Working with Maple, writing a script. In the last 2 days, i have tried several things. But i cannot find much information on the 3 norm. the 1 2 and infinity are simple. Honestly, i have nothing done and i have no idea where to start. Virtually given up. – 2012-04-28
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0It's an optimization problem with constraints... Hint: You can replace the constraint $\|x\|_3=1$ by $\|x\|_3\leq 1$ which makes it a convex constraint... – 2012-04-28