Oct 14, 2016

Matrix Norm

The class (CS532) uses as a textbook Lars Eldén – Matrix Methods in Data Mining and Pattern Recognition. Matrix norm is defined as (p.18) $$\|A\| = \sup_{x\neq0}\frac{\|Ax\|}{\|x\|}$$ I also find an equivalent definition (e.g. p.61) $$\|A\| = \sup_{\|x\|=1}\|Ax\|$$ For a while, I did not get how imposing $\|x\|=1$ is harmless, but now take it as follows. Let $d = \|x\| > 0$ and $y = \frac{x}{\|x\|} = \frac{x}{d}$, and therefore $\|y\|=1$. Then, $$ \begin{split} \|A\| &= \sup_{x\neq0}\frac{\|Ax\|}{\|x\|} \\ &= \sup_{x\neq0, \|y\|=1}\frac{\|Ayd\|}{d} \\ &= \sup_{x\neq0, \|y\|=1}\frac{d\|Ay\|}{d} \\ &= \sup_{\|y\|=1}\|Ay\| \end{split} $$

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