NsF / Forbidden Mathematics / Matrix Multiplication / NSF::20220202::2020

Please ignore IDs -- they are for robots.
Please ignore titles -- they are for me.
Please ignore errors in English for now. It would be (probably) improved.
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If you like to start discussions, it is better to do this in your blog. You can reuse IDs and images. It is better if your changes are recognizable, but it does not matter. This is a draft, and I would not explain yet the concepts of extensions.

Sometimes you discover that the modern civilization has lost some pretty obvious knowledge. You can only wonder.

It is understandable that the modern generations of mathematicians do not know functional analysis. Our professor had said that his bright PhD student had spent two months to understand the basics of his own mathematical knowledge (which he had excellently demonstrated many times) and to get the first real life equation right.

Of course, this time waste is unacceptable today. A modern PhD student feeds data and equations into a computer and in two months gets almost ready results. (Note, we live now in the world of almost ready everything.) This means, this modern PhD student performs the main task -- to demonstrate achievements -- and nobody bothers, if his equations are wrong and if the results the computer had given him, her -- or now it, them, ver, xem, etc. -- is meaningless. (By the way, if you value this modern English, it is better for you to stop and leave here. You could feel offended for mathematics you know.)

This means, the functional analysis is rightfully forgotten because it is obviously ineffective for modern science. The funny discovery of the December 2021 was the information that some people do not understand the matrix multiplication in the way it is explained to the engineers.

You multiply a 2-dimensional matrix A by a 2-dimensional matrix B. (As a side note: Your language thinks for you and it would be better to use the German "to multiple with" or the Russian "to multiple on", but we stay with the wrong correct English.)

.-			-.		.-					-.
|	a1,1,1	a1,2,1	|		|	b1,1,1	b1,1,2	b1,1,3	b1,1,4	|
|	a2,1,1	a2,21	|	×	|	b1,2,1	b1,2,2	b1,2,3	b1,2,4	|
|	a3,1,1	a3,2,1	|		*-					-*
*-			-*

The result of this multiplication is a 2-dimensional matrix C.

.-					-.
|	c1,[1,2],1	c1,[1,2],2	c1,[1,2],3	c1,[1,2],4	|
|	c2,[1,2],1	c2,[1,2],2	c2,[1,2],3	c2,[1,2],4	|
|	c3,[1,2],1	c3,[1,2],2	c3,[1,2],3	c3,[1,2],4	|
*-					-*

The equation is A×B=C. Anything is good unless you ask, why the cells in the matrix C are calculated in the following way.

c_1,1,1   =   (a_1,1,1   ×   b_1,1,1)     +     (a_1,2,1   ×   b_1,2,1)
c_2,1,1   =   (a_2,1,1   ×   b_1,1,1)     +     (a_2,2,1   ×   b_1,2,1)
...

The correct answer is: Because a result of a multiplication of two 2-dimensional matrices is 3-dimensional, but it is too complicated to work with 3-dimensional matrices.

The 3-dimensional multiplication is (or at least should be) obvious for an engineer and the explanations are banal, but it happens even golos_dobra does not understand how engineers explain the matrix multiplication. This explains why his illustrations are lacking. Below I correct some of them.

Note, there is a fundamental difference between "to know" and "to understand". The modern civilization turns from the civilization of understanding into a civilization of knowledge (or more precise: into a civilization of googled knowledge). There is also a fundamental difference between reading how to think and training oneself in the engineering thinking. At the longest night of the year 2 c.d. (COVID domini) I had drawn some explanation images for my son because this theme would be good to learn some aspects of the engineering thinking. The following text contains these images with some additional explanations.

I do not expect that everybody would understand them. I could prepare a perfect textbook for children which even a modern mathematician with the modern flat thinking could understand but it would be too boring to write. By the way, it would be not interesting to spend time for converting the notes for my son into something presentable, but I was interested in understanding how to generate SVG images and this theme was a good for a throw-away prototype. This means, all images below were generated. Some of them contain manually added decorations. The software developers create software to play with it, but at some point you should stop. It was too time consuming to write code for generation of additional decorations and too boring to modify other images manually. If you like, you could print them and draw on them the additions that could help you to obtain better understanding. If you scan the results and then destroy the paper, your process would be called "a paperless technology". (Do not be too romantic and do not let to blind you with cool buzzwords -- they all do the same thing.)

Please, do not ask about my source code. If you had read "The Software Dragons", a throw-away prototype is made to get the first outcome -- the result, and the third outcome -- the knowledge. The source code -- the second outcome of a software project -- is initially not intended to be used anymore. It is always more effective to rewrite the first draft completely on base of the obtained knowledge and the discovered top-level mistakes in the software architecture. However, today the former throw-away prototype would be called "a release". I have now the source code on the same level, but in the contrary to the modern development processes it does what it was intended to do, and it does not contain errors which must be found by users. It is simply a draft.

This text contains keys you can use to find the answers yourself. I advise to collect them and to think before you read further. (I do not expect that everybody could understand these explanations, but I expect some engineers to learn something new.) Note, this is a web post and I -- in comparison to the book pages -- cannot control what you could see. It is also too boring to write complete and precise explanations. If you are interested to know how engineers think, I advise you to take a pen and a lot of graph paper (I use 5 mm squares), and to try to find out the answers before you see them. I would use color pens for explanations for children and for people who do not understand engineering -- for instance, for mathematicians --, but all images below are for engineers, engineers can understand black and white.

( Read more... )