In the definition of the matrix norm, we have $$\|A\|=\!\!\sup_{\substack{x\in \mathbb{R}^n\\ ||x||=1}} \!\! \|Ax\|$$ I was wondering if we can replace the condition $\|x\|=1$ with $\|x\|\leq 1$. Will the two conditions give the same definition? If so, then why are we not considering the latter one as it will also give us a convex set along with being compact. Since I am new to norms, I would request you to please suggest to me some good books that can make my concepts clear. Thank you.
2026-03-28 00:48:37.1774658917
A question on altering a definition of matrix norm
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Yes, by homogeneity of the norm, you get the same definition (homogeneity: $\| t x\| = t\|x\|.$)