Category Archives: Matrix representation

Using the Cayley-Hamilton theorem to find ‘all’ multiplications in 3D space.

It is a bit vague what exactly a multiplication is, but I always use things that ‘rotate over the dimensions’. For example on the 3D complex space the imiginary unit is written as j and the powers of j simply rotate over the dimensions because:

j = (0, 1, 0)
j^2 = (0, 0, 1) and
j^3 = (-1, 0, 0). Etc, the period becomes 6 in this way because after the sixth power everything repeats.

In this post we will look at a more general formulation of what the third power of j is. The Cayley-Hamilton theorem says that you can write the third power of 3 by 3 matrices always as some linear combination of the lower powers.

That is what we do in this post; we take a look at j^3 = a + bj + cj^2. Here the a, b and c are real numbers. The allowed values that j^3 can take is what I call the ‘parameter space’. This parameter space is rather big, it is almost 3D real space but if you want the 3D Cauchy-Riemann equations to fly it has to be that a is always non zero. There is nothing mysterious about that demand of being non zero: if the constant a = 0, the imaginary unit is no longer invertible and that is the root cause of a whole lot of trouble and we want to avoid that.

It is well known that sir Hamilton tried to find the 3D complex numbers for about a full decade. Because he wanted this 3D complex number space as some extension of the complex plane, he failed in this detail and instead found the quaternions… But if the 3D numbers were some extension of the 2D complex plane, there should be at least one number X in 3D such that it squares to minus one. At the end I give a simple proof why the equation X^2 = -1 cannot be solved in 3D space for all allowed parameters. So although we have a 3D ocean of parameters and as such an infinite amount of different multiplications, none of them contains a number that squares to minus one…

I gave a small theorem covering the impossibility of solving X^2 = -1 a relative harsh name: Trashing the Hamilton approach for 3D complex numbers. This should not be viewed as some emotional statement about the Hamilton guy. It is just what it says: trashing that kind of approach…

This post is 7 pictures long, each of the usual size of 550×775 pixels.

Test picture, does jpg upload again?












Sorry for the test picture, but the seven jpg pictures refused to upload. And that is strange because they are just seven clean jpg’s. Now it is repaired although I do not understand this strange error.

Anyway have a cool summer. Till updates.

The Cayley-Hamilton theorem neglected for 25 years?

That is strange, if you don’t know the Cayley-Hamilton theorem; it is the finding that every square matrix A, if you calculate the characteristic polynomial for the matrix A it is always zero. At first this is a very surprising result, but it is easy to prove. It’s importance lies in the fact that in this way you can always break down higher powers of the matrix A in lower powers. In the study of higher dimensional complex and circular numbers we do this all the time. If in 3D space I say that the third power of the imaginary component is minus one, j^3 = -1, we only write the third power as a multiple of the zero’th power…

In this post I will give two simple proofs of the Cayley-Hamilton theorem and although in my brain this is just a one line proof, if you write it down it always gets longer than anticipated.

At the end I show you an old video from the year 1986 from the London Mathematical Society where it is claimed that the CH theorem was neglected for 25 years. Now Hamilton is also famous for having sought the 3D complex numbers for about a full decade before he gave up. And I still do not understand why Hamilton tried this for so long but likely he wanted to include the imaginary unit i from the complex plane in it and that is impossible. Or may be he wanted a 3D complex number system that is also a field (in a field all elements or numbers that are non-zero have an inverse, in algebra wordings; there are no divisors of zero). A 3D field is also impossible and in this post I included a small proof for that.

Furthermore in this post at some point may be you read the words ‘total incompetents’ and ‘local university’. You must not view that as some emotional wording, on the contrary it is a cold clinical description of how math goes over there. So you must not think I am some kind of frustrated person, for me it is enough that I know how for example to craft a 3D complex number system. If they don’t want to do that, be my guest. After all this is a free country and we also have this concept of ‘academic freedom’ where the high shot math professors can do what they want.

And what is this ‘academic freedom’ anyway? If for example unpaired electrons are never magnetically neutral but electron pairs always are magnetically neutral, can the physical reality be that electrons are magnetic dipoles? Of course not, that is a crazy idea to begin with. But 97 years of academic freedom since the Stern-Gerlach experiment have never ever brought any meaningful understanding of the magnetic properties of the electron. If it acts as a magnetic charge and you say it is not a charge it is easy to understand how you can fool yourself for about one century of time.

This post is seven pictures long although the last picture is empty.
The two proofs of the Cayley/Hamilton theorem is how I would prove such a thing but good theorems always have many proofs. All pictures are of the size 550×775 pixels.













Why is the seventh picture without math?

Here is the old video from 1986 where it is claimed the Cayley-Hamilton theorem was neglected for about 25 years. Oh oh oh what a deep crime. But the human mind is not made to produce or understand math, so in my view 25 years is a short period of time if in the good old days math professors were equally smart as the present day math professors. The title of the video is The Rise and Fall of Matrices.

Matrices saved my life from crazy math professors.

Ok let me leave it with that an not post a link to the top wiki on the Cayley-Hamilton theorem where all kinds of interesting proofs are given. Till updates my dear reader.

Part 18: Calculating the 4D number tau it’s inverse in a very simple way.

It is the shortest day of the year today and weirdly enough I like this kind of wether better compared to the extreme heat of last summer. Normally I dislike those long dark days but after so much heat for so long I just don’t mind the darkness and the tiny amounts of cold.

In the previous post we found a general way of finding all inverses possible in the space of the 4D complex numbers. Furthermore in the post with the new Cauchy integral representation we had to make heavy use of 1/8tau and as such it is finally time to look at what the inverse of tau actually is.

I found a very simple way of calculating the inverse of the number tau. It boils down to solving a system of two linear equations in two variables. As far as I know reality, most math professionals can actually do this. Ok ok, for the calculation to be that simple you first must assume that the inverse ‘looks like’ the number tau in the sense it has no real component and it is just like tau a linear combination of those so called ‘imitators of i‘.

This is a short post, only five pictures long. I started the 4D complex number stuff somewhere in April of this year so it is only 8 month down the timeline that we look at the 4D complex numbers. It is interesting to compare the behavior of the average math professor to back in the time to Hamilton who found the 4D quaternions.

Hamilton became sir Hamilton rather soon (although I do not know why he became a noble man) and what do I get? Only silence year in year out. You see the difference between present day and past centuries is the highly inflated ego of the present day university professors. Being humble is not something they are good at…

After having said that, here are the five pictures:

All in all I have begun linking the 4D complex numbers more and more in the last 8 months. On details the 4D complex numbers are very different compared to 3D and say 5D complex numbers but there are always reasons for that. For example the number tau has an inverse in the space of 4D complex numbers but this is not the case in 3 or 5D complex numbers space.

Well, have a nice Christmas & likely see you in the next year 2019.

Part 17: The inverse of a 4D complex number old school style (via minor matrices).

Ha, a couple of weeks back I met an old colleague and it was nice to see him. We made a bit of small talk and more or less all of a sudden he said: ‘But you still can always do this’. And he meant getting a PhD in math. 

I was a bit surprised he did bring this up, for me that was a station passed long ago. But he made me thinking a bit, why am I not interested in getting a math degree? 

And when I thought it out I also had to laugh: Those people cannot go beyond the complex plane for let’s say 250 years. And the only people I know of that have studied complex numbers beyond the complex plane are all non-math people. Furthermore inside math there is that cultural thing that more or less says that if you try to find complex numbers beyond the complex plane, you must have a ‘mental thing’ because have you never heard of the 2-4-8 theorem? 

Beside this, if I tried it in the years 1990 and 1991 with very simple: Here this is how the 3D Cauchy Riemann equations look… And you look them in the eyes, but there is nothing happening behind those eyes or in the brain of that particular math professor. Why the hell should I return and under the perfect guidance of such a person get a PhD? 

I am not a masochist. If complex numbers beyond the complex plane are ignored, why try to change this? After all this is a free world and most societies run best when people can do what they are good at. Apparently math like I make simply falls off the radar screen, I do not have much problems with that. 

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After having said that, this update Part 17 in the basics to the 4D complex numbers is as boring as possible. Just finding the inverse of a matrix just like in linear algebra with the method of minor matrices.

Believe me it is boring as hell. And after all that boring stuff only one small glimmer of light via crafting a very simple factorization of the determinant inside the 4D complex numbers. So that is very different from the previous factorization where we multiplied the four eigenvalue functions. From the math point it is a shallow result because it is so easy to find but when before your very own eyes you see the determinant arising from those calculations, it is just beautiful. And may be we should be striving a tiny bit more upon mathematical beauty…

This post is nine pictures long in the usual size of 550×775 pixels.

 

As an antidote against so much polynomials like det(Z), with 2 dimensions like a flatscreen television, you can do a lot of fun too. The antidote is a video from the standupmaths guy, it is very funny and has the title ‘Infinite DVD unboxing video: Festival of the Spoken Nerd’. Here is the vid:

End of this update, see you around.

Ok ok, a few days later I decided to write a small appendix to this post and in order too keep it simple let’s calculate the determinant of the 3D circular numbers. I have to admit this is shallow math but despite being shallow it gives a crazy way to calculate the determinant of a 3D circular number…

So a small appendix, here it is:

And now you are really at the end of this post.

In the next post let’s calculate the inverse of the 4D complex tau number. After all a few months back I gave you the new Cauchy integral representation and I only showed that the determinant of tau was nonzero.

But the fact that the Cauchy integral representation is so easy to craft on the 4D complex numbers arises from the fact the inverse of tau exists in the first place. In 3D the number tau is not invertible, and Cauchy integral representations are much more harder to find.

Ok, drink a green tea or pop up a fresh pint, till updates.

Calculation of the 4D number tau diagonal matrix style.

In the begin of this series on basic and elementary calculations you can do with 4D complex numbers we already found what the number tau is. We used stuff like the pull back map… But you can do it also with the method from the previous post about how to find the matrix representation for any 4D complex number Z given the eigenvalues.

Finding the correct eigenvalues for tau is rather subtle, you must respect the behavior of the logarithm function in higher dimensions. It is not as easy as on the real line where you simply have log ab = log a + log b for positive reals a and b.

But let me keep this post short and stop all the blah blah.

Just two nice pictures is all to do the calculation of the 4D complex number tau:

 

(Oops, two days later I repaired a silly typo where I did forget one minus sign. It was just a dumb typo that likely did not lead to much confusion. So I will not take it in the ‘Corrections’ categorie on this website that I use for more or less more significant repairs…)

Ok, that was it.

Diagonal matrices for all 4D complex numbers.

This website is now about 3 years old, the first post was on 14 Nov 2015 and today I hang in with post number 100. That is a nice round number and this post is part 15 in the series known as the Basics for 4D complex numbers.

We are going to diagonalize all those matrix representations M(Z) we have for all 4D complex numbers Z. As a reader you are supposed to know what diagonalization of a matrix actually is, that is in most linear algebra courses so it is widely spread knowledge in the population.

Now at the end of this nine pictures long post you can find how you can calculate the matrix representation for M(l) where l is the first imaginary unit in the 4D complex number system. And I understand that people will ask full of bewilderment, why do this in such a difficult way? That is a good question, but look a bit of the first parts where I gave some examples about how to calculate the number tau that was defined as log l. And one way of doing that was using the pull back map but with matrix diagonalization you have a general method that works in all dimensions.

Beside that this is an all inclusive approach when it comes to the dimension, in practice you can rely on internet applets that use commonly known linear algebra. Now if you are a computer programmer you can automate the process of diagonalization of a matrix. I am very bad in writing computer programs, but if you can write code in an environment where you can do symbolic calculus in your code, it would be handy if that is on such a level you can use the so called roots of unity from the complex plane. After all the eigenvalues you encounter in the 4D complex number system are always based on these roots of unity and the eigenvectors are too…

This post number 100 is 9 pictures long, as usual picture size is 550 x 775 pixels.
In the next post number 101 we will use this method to calculate the matrix representation of the number tau (that is the log of the first imaginary unit l).

Ok, here are the pictures:

That´s it, in the next post we go further with the number tau and from the eigenvalues of tau calculate the matrix representation. So see you around.

Part 14: The Cauchy integral representation for the 4D complex numbers.

It took me longer than expected to craft this update. That is also the nature of the subject; you can view and do math with Cauchy integral representation in many ways. In the end I settled on doing it just for polynomials of finite degree and even more simple: these polynomials are real valued on the real line. (So they have only real coefficients and after that are extended to the space of 4D complex numbers).

In another development, last week we had the yearly circus of Noble prizes and definitely the most cute thing ever was those evolving protein molecules. Because if you can use stuff like the e-coli bacteria you can indeed try if you can (forcefully) evolve the proteins they make… That was like WOW. Later I observed an interview with that chemistry Nobel prize winner and she stated that when she began she was told ‘gentlemen don’t do this kind of thing’.

So she neglected that ‘gentlemen stuff’ and just went on with it. That is a wise thing because if you only do what all those middle age men tell you to do you will find yourself in the very same hole as they are in…
The physics prize was also interesting, for myself speaking I was glad we did not observe those physics men totally not understanding electron spin but with the usual flair of total arrogance keep on talking about spin up and spin down.

You can also turn that spin nonsense upside down: If elementary particles only carry monopole electrical charge than why should electrons be bipolar when it comes to magnetism? That Gauss law of magnetism is only a thing for macroscopic things, there is no experimental proof it holds for quantum particles…

But let’s talk math because this update is not about what I think of electron spin. This is the second Cauchy integral representation I crafted in my life. Now the last years I produced a whole lot of math, my main file is now about 600 pages long. But only that very first Cauchy integral representation is something that I printed out on a beautiful glossy paper of size A0. That first Cauchy integral representation was on the space of 3D numbers and there life is hard: The number tau has determinant zero and as such it is not invertible. But I was able to complexify the 3D circular numbers and it was stunning to understand the number tau in that complexification of the 3D circular numbers. Just stunning…

Therefore I took so much time in trying to find an easy class of functions on the space of 4D complex numbers. I settled for easy to understand polynomials, after all any polynomial gives the same value everywhere if you write them as a Taylor series.
Since this property of polynomials is widely spread I can safely say this in this part 14 of the basics to the 4D complex numbers we have the next Theorem:
THEOREM: The math will do the talking.
PROOF: Just read the next 12 pictures. QED.

As usual all pictures are 550 x 775 pixels in size. I also use a thing I name ‘the heart of the Cauchy integral’, that is not a widely known thing so take your time so that the mathematical parts of your brain can digest it…

I truly hope the math in this update was shallow enough so you can use it in your own path of the math that you like to explore.

End of this post, may be in Part 15 we will finally do a bit more about the diagonalization of 4D complex numbers because that is also a universal way of finding those numbers tau in the different dimensions like the 17D circular numbers & all those other spaces.

Have a nice life or try to get one.

Part 13 of the basics of 4D complex numbers: Factorization of the determinant.

It is about high time for a small update. Originally I wanted to include a little rant against all those math professors that have stated that the so called Euler formulae for the exponential circle in the complex plane is the most beautiful piece of math ever.

How can you say that and after all those other exponential circles and curves I found stay silent year in year out? We now have a fresh academic year and likely the new year nothing will happen again.

The same goes for magnetism, if it is true electrons carry magnetic charge one way or the other this will have huge economical impacts in the long run. Not only can you better understand how spintronic devices can work but as a negative also understand how nuclear fusion with those Tokomak things will never work because of the electron acceleration.

In my view this is important given the speed of climate change and how slow we react on it while at the same time we are always promised golden mountains of almost free energy if only we had nuclear fusion…

But likely university people are university people so we will just observe one more academic year of just nothing.

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After having said that, in this post we calculate the determinant via multiplying the four eigenvalues every 4D complex number Z has. Of course that is always when such a Z is viewed as it’s matrix representation M(Z).

You can do much more with that kind of stuff, in the previous part number 12 we unearthed the eigenvalues and eigenvectors so we can also do the nice thing of matrix diagonalization. I have not planned anything about the content in the next few parts of these small series on 4D complex numbers. So we’ll see.

This post is five pictures long, at the end I also show you the so called cylinder equation from 3D space (circular version).

Have fun reading it and thinking about it a little bit. See you around.

Before the actual post I will show you the teaser picture as published on the other website, it contains the matrix who’s determinant we are going to factor. When I read all these posts on 4D complex numbers backwards it was only in the basics number 3 where I mentioned this explicit matrix representation M(Z).

In the teaser picture you also see the main result. I never worked out the determinant of the matrix M(Z) via a method like expansion via the minors or so. Just going straightforward for the eigenvalue functions and multiply these in order to get the determinant for any 4D complex number Z.

So it is the determinant of the matrix representation M(Z) we are going to factorize.
Here we go:

 

 

 

 

 

Ok, let´s not rant because why waste all that emotional energy on university people?
But it is now almost your years back since I crafted a pdf about the first 10 exponential circles and curves I found:

An overview of exponential circles and curves in …
http://kinkytshirts.nl/pdfs/10_exponential_circles_and_curves.pdf

That was from 22 December 2014. We can safely conclude that at most universities a lot is happening, but it is mostly weird stuff. Weird stuff like ‘The sum of all integers equals minus 1/12…’ Oh oh, if you have people like that inside your ranks how can that bring any good?

So is it science or comedy in the next video?

It is more like comedy I just guess…

But what would life be without comedy? That would also be a strange place to live, a life without observable comedians likely is a less funny place to live in.

Hey let’s pop open one more pint of beer. By the way in the quantum world every thing is different and at first I could not believe that, why would that be? It was years and years later I found out that quantum particles like electrons never drink beer. Just never. And at that point in time I finally understood just how different the quantum world is compared to our human world.

Till updates.

Part 12 of the 4D complex numbers: The four eigenvalue functions for a arbitrary 4D complex number.

In Part 11 we found the eigenvalues and eigenvectors of the first imaginary unit l (with of course the property l^4 = -1). But if we have those eigenvalues for l,it is easy to find the eigenvectors of powers of l.

But every 4D complex number is a sum of a real part and the three imaginary parts with the units  l,  l ^2 and l ^3. So from a linear algebra point of view it is also easy to find the eigenvalues of such combinations of  l. Needless to say that in this post you should read  l and it powers mostly as it’s matrix representation M(l).

I have to admit that I very often avoid the constant need for the matrix representation, for example I mostly write det(l) for the determinant of M(l). But it makes texts just so unreadable if you constant write det(M(l)), I do not like that.

When you understand how to calculate the eigenvalue functions as shown below, please remark it looks a like the so called discrete Fourier transform.

This post is five pictures long (each 550×775 pixels) although the last picture is rather empty… And why no an empty picture? After all when you had a paper book in the good old days, there were always empty pages in it. Sometimes it was even written that ‘This page is left empty intentionally’. And in those long lost years that was the crime of the century because if you write on a page that it is empty, that is never true…

In the next post we will take a look at the missing equation we still have when it comes to the calculation of the 4D exponential curve when it comes to the  sphere/cone equation. we missed one equation to arrive at a 1D solution for our exponential curve.

If you also include the demand that the determinant of the exponential curve is 1 all of the time, you can squeeze out more equations inside the 4D complex numbers.

Before we say goodbye, here is a link to the matrix representation of the discrete Fourier transform. But take you time when thinking about matrices like in the link or the Omega matrix as above.

DFT matrix
https://en.wikipedia.org/wiki/DFT_matrix

Ok, that was it. Till updates.

Part 11 of the 4D complex numbers: The eigenvalues and eigenvectors from the first imaginary unit.

First a short magnetic update:

In reason 66 as why electrons cannot be magnetic dipoles I tried to find a lower bound for the sideway acceleration the electrons have in the simple television experiment.

To put it simple: How much sideway acceleration must the electrons have to explain the dark spots on the screen where no electrons land?

The answer is amazing at first sight: about 2.5 times 10^15 m/sec^2.
This acceleration lasts only at most two nano seconds and in the end the minimum sideway speed is about 5000 km/sec so while the acceleration is such a giant number it does not break relativity rules or so…
Here is the link:

16 Aug 2018: Reason 66: Side-way electron acceleration as in the television experiment.
http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#16Aug2018

You know I took all kinds of assurances that it is only a lower bound on the actual acceleration that takes place. For example I took the maximal sideway distance as only 0.5 cm. Here is a photo that shows a far bigger black spot where no electrons land, so the actual sideway distance if definitely more than 0.5 cm.


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The math part of this post is not extremely thick in the sense you can find the results for yourself with the applet as shown below. Or by pencil & paper find some 4D eigenvalues and the corresponding eigenvectors for yourself.

But we need them in order to craft the so called eigenvalue functions and also for the diagonal matrices that come along with all of the matrix representations of the 4D complex numbers Z.

I hope I wrote it down pretty straightforward, this post is five pictures long. And if you like these kind of mathematical little puzzles: Try, given one of the eigenvalues like omega or omega^3, find such an eigenvector for yourself. It is really cute to write them down, multiply them by the eigenvalue and observe with your own eyes that indeed we have all that rotation over the dimensions included that omega^4 = -1 behavior.

This post is five pictures long, it is all rather basic I hope.

The applet used is from the WIMS server (https://wims.sesamath.net/), look for the Matrix calculator in the section on Online calculators and plotters.

For the time being I think that in Part 12 we will craft the eigenvalue functions for any 4D complex number Z. Ok, that was it for this update.