I have a theorem in my book on matrix computations that states the following:
A vector norm and its induced matrix norm satisfy the inequality:
$\|Ax\|\leq \|A\|$$\|x\|$ where A $\in R^{nxn}$ and x $\in R^n$.
The book states that equality holds, "if and only if x is a vector for which the maximum magnification is attained. (That such a vector exists is actually not obvious. It follows from a compactness argument that works because $R^n$ is a finite-dimensional space. We omit the argument.)"
I am interested in knowing how I would go about showing that equality holds if x is a vector for which max magnification is attained. This is not a homework problem; however, I would like to know for my upcoming exam.
Best Answer
In $\mathbb{R}^n$, the closed unit ball $\overline{B}$ is compact. The function $f(x) = \|Ax\|$ is continuous, and continuous functions attain their maximum on a compact set, ie, there exists $x_0 \in \overline{B}$ such that $f(x_0) = \max_{x \in \overline{B}} f(x)$. In particular, $\|A x_0 \| = \max_{x \in \overline{B}} \|Ax\|$.
The induced norm is defined as $\|A\| = \max_{x \in \overline{B}} \|A x\|$, and the above remark shows that this is attained for some $x_0$.