Oct 17, 2016

Let People Pay

I always believe that people want to do right things and look for right ways to expend their resources — money, time, kindness, attention, or any physical and cognitive assets. So, when it comes to sustainability of economy and society at large, as Amanda Palmer demonstrated it, the question is not “How do we make people pay?” but rather “How do we let people pay?”

Rightness is idiosyncratic originating from ethical, cultural and social principles. In addition, as these factors interact and coevolve with each other, rightness is also dynamic. Even seemingly ubiquitous values like trust could considerably differ across people. So, we may not see the grand general social theory in the near future. Economics I would like to pursue is humbly ad hoc but recognizes such idiosyncrasy, not as fringe, but as fundamental, and designs systems where people feel right and willing to expend their resources. I believe it would be a meaningful progress in social science and, this way, society would be a bit more sustainable and a bit more pleasant.

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, $$ \begin{split} \|A\| &= \sup_{x\neq0}\frac{\|Ax\|}{\|x\|} \\ &= \sup_{x\neq0, \|y\|=1}\frac{\|Ayd\|}{d} \\ &= \sup_{x\neq0, \|y\|=1}\frac{d\|Ay\|}{d} \\ &= \sup_{\|y\|=1}\|Ay\| \end{split} $$