User talk:Eliosh

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Thank you for your fixes at Cauchy-Schwarz inequality. Just one remark. The proof over there seems to be done in the case of the real scalar product. In this case the order in  does not matter. If you want to make the proof work for complex scalar products, there is more work to be done. Otherwise, I think one needs to explictely mention there that the proof only works for real scalar products. What do you think? Oleg Alexandrov 18:02, 4 Apr 2005 (UTC)

Hi, The change was done to match the definiton of inner product as it is given here in Wikipedia (which is complex). 16:30, 6 Apr 2005 (UTC)


 * Right. The theorem was stated for complex inner product, but the proof was done (probably by somebody else) only for the real product. And what you modified was the proof. So, my humble suggestion was that since you started modifying the proof to make it work for the complex case, you could as well modify it completely, as the way the proof is now, it still assumed real inner prodcut in many places.


 * No, you did not do anything wrong. Relax. :) Oleg Alexandrov 16:57, 6 Apr 2005 (UTC)

Cauchy-Schwarz inequality
Dear Eliosh. I think I mentioned this before. The proof at Cauchy-Schwarz inequality as written, does not work for the complex inner product. That proof assumes the scalar product of two vectors is a real number. Of course the proof can be modified, but you cannot claim the proof is for the complex case while it is not. Thanks, Oleg Alexandrov 5 July 2005 16:05 (UTC)

Oleg, may be I misuderstood you. This proof uses, for example the fact that  = * (complex conjugate) and thus it assumes that  is complex. Obviously it is true for the real case.

Hi Eliosh. I am talking about the paragraph:

Proof

 * Real inner product spaces

To prove this inequality note it is trivial in the case y = 0. Thus we may assume  is nonzero. Further, let $$ \lambda \in \mathbb{R} $$. Thus we may let


 * $$ 0 \leq \langle x-\lambda y,x-\lambda y \rangle$$


 * which equals


 * $$ = \langle x-\lambda y,x \rangle - \lambda \langle x-\lambda y,y \rangle. $$


 * We now choose $$ \lambda = \langle y,x \rangle \cdot \|y\|^{-2}. $$


 * Plugging in for $$ \lambda $$ we wind up getting


 * $$ 0 \leq \|x\| ^2 - \langle x,y \rangle^2 \cdot \|y\|^{-2}$$


 * which is true $$ \iff $$


 * $$ \langle x,y \rangle^2 \leq \|x\|^2 \|y\|^2. $$


 * Taking the square root gives us


 * $$ \big| \langle x,y \rangle \big| \leq \|x\| \|y\| $$ Q.E.D.

from Cauchy-Schwarz inequality. I see here $$ \lambda \in \mathbb{R} $$ which equals
 * $$ \lambda = \langle y,x \rangle \cdot \|y\|^{-2}, $$ but which is not real.

I also see $$ \langle x,y \rangle^2$$ which has to be an absolute value if it is a complex number. So, I think a bit more of work is needed before this proof would work for complex products. But not much work is needed. I should have done it myself rather than pestering you I guess. These days I don't have the time, but I plan to get to it soon. Cheers, Oleg Alexandrov 13:22, 11 July 2005 (UTC)

Oleg, you are absolutely right, however I think these errors are rather typos, not real errors. I'll fix it right now. --Eliosh.


 * I did not say they are errors, I said that the proof did not yet apply for complex numbers. Now it does.:) Thank you for your fixes. I am sorry I made so much fuss about it. Oleg Alexandrov 16:32, 12 July 2005 (UTC)

QR decomposition
I saw that you generalized the definition in QR decomposition. This is of course a good thing and I am grateful for it. Unfortunately, there are some aspects that I am less happy with it. Since you seem not to have that much experience here, I thought it would be better to share my concerns rather than change your contribution immediately.

We try to make our articles accessible to as many readers as possible. One way to achieve this is to start with a simple definition and to introduce the more general definition later. In this case, I would prefer if at the top we restrict ourselves real square matrices, and put the definition for complex rectangular matrices further down.

Two minor gripes: Try not to use tags in the text and list wherever you found the definition as a reference. You can read all this in more detail at How to write a Wikipedia article on mathematics. You may also be interested in WikiProject Mathematics, which is just a bunch of people who like to contribute to the maths articles here.

Most importantly, I hope that you continue your work here. Cheers, Jitse Niesen (talk) 11:33, 19 July 2005 (UTC)

Hi. Of course there is always a question how many details/simplifications should be in your article. I think complex numbers are not more difficult than real ones. Moreover, people interested in QR decomposition should be familar with complex numbers. Eli Osherovich 18:08, 20 July 2005 (UTC)