Question

# 3. Estimates for vector and matrix norms, [4+3+3pt (extra credit)] 'There is no strict bound above which a system becomes poorly conditioned-but one usually considers conditionsnumbers large when they are

> 10°, 10´,... or even larger.2The commands needed in MATLAB are abs and sum. Most commands can not only applied to numbers, but also tovectors, where they apply to each component.3In MATLAB use the stop watch commands tic and toc.*Use help norm to find out how to obtain the matrix norm that is induced by either the 1,2 or oo-vector norm. (a) For a vector v E R", do the following definitions N : R" → R defined norms? If yes, prove that they satisfy the axioms. If no, argue why (e.g., giving an example that shows that the axioms are not satisfied). i. N(v) := |vi|, where n > 1. ii. N(v) := #{i : vi != 0}, i.e., the number of nonzero entries in the vector. iii. N(v) :=||v||1+ ||v||oo. iv. N(v) := ||v||2. (b) Show that, for any v E R^n, we have In each case, give an example of a nonzero v for which equality is obtained. (c) (extra credit) Let us generalize the definition of matrix norms to non-square matrices. We define the || · ||, matrix norms (p E {1,2,oo}) for an m x n matrix A by \|A\|_{p}=\sup _{v \in \mathbb{R}^{n} \backslash\{0\}} \frac{\|A v\|_{p}}{\|v\|_{p}} where the norm in the numerator is defined on R^m and the norm in the denominator is defined on R^n.Using the problem above, show that \|A\|_{\infty} \leq \sqrt{n}\|A\|_{2} \quad \text { and } \quad\|A\|_{2} \leq \sqrt{m}\|A\|_{\infty} In each case, give an example of a nonzero matrix A for which equality is obtained. \|\boldsymbol{v}\|_{\infty} \leq\|\boldsymbol{v}\|_{2} \quad \text { and } \quad\|\boldsymbol{v}\|_{2}^{2} \leq\|\boldsymbol{v}\|_{1}\|\boldsymbol{v}\|_{\infty}

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