Can someone explain why the term in the first {}
equals the second term in the {}
:
Why the max can be found only with normalized vectors?
2
$\begingroup$
matrices
vector-spaces
norm
2 Answers
1
What is the norm of $\frac{x}{\| x \|}$? Then use the scalar-linearity of the mapping $x\mapsto Ax$.
1
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 $A(kx)>kA(x)$.)
-
0@0x90 http://en.wikipedia.org/wiki/Axiom thank me later ~ – 2012-12-22
-
0http://www.caam.rice.edu/~embree/caam453/lecture2.pdf – 2012-12-22
-
0@0x90 what about that? Just some painfully inprecise reading script? – 2012-12-22
-
0That is a little bit more relevant in compare to your link into wiki Axiom ... that explains my question :) thanks – 2012-12-22
-
0Dont forget to mark your question as answered ;) – 2012-12-22