# Why the max can be found only with normalized vectors?

Can someone explain why the term in the first {} equals the second term in the {} : ## 2 Answers

What is the norm of $\frac{x}{\| x \|}$? Then use the scalar-linearity of the mapping $x\mapsto Ax$.

For every $x\in K^n$ there is a normalized vector $\frac{x}{\|x\|}$; using the axioms for matrixnorms we have $\|A\frac{x}{\|x\|}\|=\frac{\|Ax\|}{\|x\|}$

Therefore, it is sufficient to only consider the maximum over vectors of norm 1. (As a matrix represents a linear function, there cannot be a $k\in K$, for which $A(kx)>kA(x)$.)

• @0x90 en.wikipedia.org/wiki/Axiom thank me later ~ – CBenni Dec 22 '12 at 13:04
• caam.rice.edu/~embree/caam453/lecture2.pdf – 0x90 Dec 22 '12 at 13:15
• @0x90 what about that? Just some painfully inprecise reading script? – CBenni Dec 22 '12 at 13:27
• That is a little bit more relevant in compare to your link into wiki Axiom ... that explains my question :) thanks – 0x90 Dec 22 '12 at 13:35
• Dont forget to mark your question as answered ;) – CBenni Dec 22 '12 at 13:46