Suppose $A\inℝ^{n,n}$. We Define
$$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$
Show it is a norm.
Any thoughts?
2026-04-05 22:03:51.1775426631
Norm of a Vector
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The first three properties of a norm should be very easy to verify. The one that is giving you grief is triangle inequality, but it is no more difficult than you think it is since
$$\underset{\|x\| = 1}{\sup} \left \| (A_1 + A_2)x \right \| = \sup_{\| x \| =1} \| A_1 x+ A_2x \| \leq \sup_{\|x \| = 1} \| A_1 x \| + \sup_{\| x \| = 1} \| A_2 x\|.$$