Category Archives: Uncategorized

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.

The logarithm of all 2D circular numbers (the split complex numbers).

Yesterday I was editing the six pictures for this update and all of a sudden I realized I had made a dumb dumb mistake: The pictures count down from number 7 to number 2…

I had processed them in the wrong order; I had made seven background pictures but I filled in the math text in the wrong order.

All in all I decided to leave it this way; it might be a stupid mistake but it is not a critical mistake like making a critical math error or having wrong ideas about what is actually happening on the math level. It is just an editing error and also funny. So I leave it this way.

In this post we look at the so called split complex numbers, they are the cousin of the numbers from the complex plane. The only difference is that where in the complex plane the square of the imaginary unit equals minus one, for the split complex numbers this equals plus one.

Although this is a minor change, split complex numbers are not a field because it contains non-invertible numbers outside the number 0. All I do in this post is finding the eigenvalues and eigenvectors of all split complex numbers and via taking the log of the eigenvalues we calculate what the log of an arbitrary split complex number is.

In the speak of this website the split complex numbers are just the 2D circular numbers. Remeber in all dimensions numbers are complex or circular depending if the first imaginary unit equals -1 or +1. You can find many more ways of crafting a multiplication but the best math results are always found in the complex and circular version of the numbers in that particular dimension…













Ok, in this post I left all things out that talks about the 4D hybrid space that is a mixture of the 2D circular and complex numbers. But as you see on inspection of the above six pictures, the eigenvalues might be always real but they can be negative. As such always pay attention when you apply that function named the log…

That was it for this post, at this point in time I have no idea what the next post will be about. After all we had this long rout of over 20 posts on the 4D complex numbers and I left a whole lot of other stuff out in that period. Stuff like 3D Gauss integers or a general definition for integration that works in all dimensions. Till updates my dear reader.

And life? Life will go on.

Originally I planned on showing you some numerical results from the circular 4D numbers while explaining there is also a number alpha in 4D. For me that would be a nice holiday away from all that 4D complex number stuff from the last months…

But the numerical applet did not work, it is still dead in the water:

http://calculator.vhex.net/calculator/linear-algebra/matrix-exponential-using-the-pade-approximation

Ok ok, I could have done those numerical showings also in rigid analysis but I guessed that calculating a 4D tau for circular numbers via analysis was too much. And I settled for a much more easy to understand thing:

The logarithmic function for every 2D circular number. In the field of professional math professors the 2D circular numbers are known as the split complex numbers.

So that is what the next post will be about: Finding log(z) for all invertible split complex numbers.

I only wrote one previous post on the 2D circular aka split complex numbers and that dates back to Nov 24 of the year 2016:

The second hybrid: a 4D mix of the complex and the circular plane.

Ha ha, now I can laugh about it but back in the time it was some hefty pain. Anyway to make a long story short: In that old post from 2016 I calculated the log for just one split complex number namely the first imaginary unit j.

Let me show you my favorite part of that old post from 2016:

So the next update will only contain 2×2 size matrices while I skip the detail that the log lives mainly in the hybrid number system from the old post.

Till updates.

The inverse theorem of Pythagoras (part 2).

Somewhere last year I just looked some nice video from the Mathologer about the theorem of Pythagoras. And since I myself have found a proof for the general theorem of Pythagoras in higher dimensions, I was puzzled about what the so called ‘inverse theorem of Pythagoras’ actually was.

Could I do that too in my general proof? And the answer was yes, but when I wrote that old proof of the general theorem of Pythagoras it was just a technical blip not worthwhile mentioning because it was a simple consequence of how those normal vectors work.

Anyway to make a long story short, a few days back I likely had nothing better to do and for some reason I did an internet search for ‘the inverse theorem of Pythagoras’. All I wanted to do is read a bit more about that from other people.

To my surprise my own writing popped up as search result number 3, that was weird because I wanted to read stuff written by other people… Here is a screenshot of the answers as given by the Google search machine:

Ok ok, not bad at search result number 3.

Now why bring this up? Well originally I forgot to post to the video that started my thinking in the first place. It is from the Mathologer and here at 16.00 minutes into his video is where my mind started to drift off:

The video from the Mathologer is here (title Visualizing Pythagoras: ultimate proofs and crazy contortions):

It is a very good video, my compliments.

After so much advertisements for the Mathologer, just a tiny advertisement for what I wrote on the subject of the inverse theorem of Pythagoras on March 20 in the year 2018:

What is the inverse Pythagoras theorem?

Ok, that was it. Till updates.

Just a short video on the Fourier stuff.

This is the shortest post ever written on this website.

I found one of those video’s where the Fourier series is explained as the summation of a bunch of circles. Likely when you visit a website like this one, you already know how to craft a Fourier series of some real valued function on a finite domain.

You can enjoy a perfect visualization of that in the video below:

Only one small screen shot from the video:

Oh oh, the word count counter says 80+ words. Let me stop typing silly words because that would destroy my goal of the ‘shortest post ever’. Till updates.

What is the inverse Pythagoras theorem?

It is already late in the evening, actually it is past midnight so I will keep the text of this post short. It was a nice day today and this evening I brewed the 23-th batch of a beer known as ‘Spin half beer’. (I name it that way because it contains only half of the dark malts I use in the beer known as dark matter…;) so it has nothing to do with electrons).

This is a very basic post about some ‘inverse Pythagoras theorem’ as came flying by in some math video. I was rather surprised that I have not seen it before but there are so many theorems out there using that old fashioned Euclidian geometry that I might have forgetten all about it.

Within 10 minutes I had a good proof for the 2D version of this ‘inverse Pythagoras theorem’. You can find it in the first picture below.

One day later when I was riding a bit around I tried to find the higher dimensional analog of that easy to understand 2D statement or theorem. And as such it crossed my mind the important role a distance number d played in my proof for the general theorem of Pythagoras that acts on simplexes that are the higer dimensional analog of 2D triangles.

Coming home it was easy to write out the details, but for me it was all so simple that does this stuff deserve the title ‘theorem’? Well make up your own mind about that, but if it is not a real complicated theorem it is still a nice and cute result…

This post is six pictures long (all 550×775 pixels beside the last one that needed a bit expansion because the math did not fit properly so that one is 600×775 pixels).

At times it might look difficult but this is only because it is in a general setting when it comes to the number of dimensions, the basic idea’s are all simple things like taking an inner product with a normalized normal vector.

Here are the six pictures:

That is a cute result but for me the normal vector is just as cute but only a bit harder to write out because that part deals with general setting where the dimension n is not fixed.

For the time being is this the end of this post. See you around my dear reader.

__________

Addendum added on 30 March 2018: In the previous post I forgot to place a link to the proof of the general theorem of Pythagoras as I crafted it once a long time ago.

Before this link I would like to show you once more how to prove the general theorem of Pythagoras for the 3D case using only the 2D theorem.

After all, that is the first basic step in my proof for the general theorem of Pythagoras…

Here are the two addendum pictures outlining how this basic step from the two dimensional plane to the 3D space goes:

Here is the link to the proof of the general theorem of Pythagoras:

The general theorem of Pythagoras (second and final post).

The general theorem of Pythagoras (second and final post).

Ok that was it, till updates.

Oops; CERN did not find magnetic monopoles.

It has to be remarked that the physics folks are very persistant to keep on trying to find the so called Dirac monopole. How this has come to be is still a miracle to me. After all if the electron has one electric charge and for the rest it is a magnetic dipole, it would look naturally to look for a particle that is a magnetic monopole and an electric dipole at the same time…

But I have never heard about such an investigation, it is only the Dirac magnetic monople and that’s it.

Here is a quote from sciencenews dot org:

If even a single magnetic monopole were detected, the discovery would rejigger the foundations of physics. The equations governing electricity and magnetism are mirror images of one another, but there’s one major difference between the two phenomena. Protons and electrons carry positive and negative electric charges, respectively, but no known particle has a magnetic charge. A magnetic monopole would be the first, and if one were discovered, electricity and magnetism would finally be on equal footing.

Source:

Magnets with a single pole are still giving physicists the slip
https://www.sciencenews.org/article/magnetic-monopoles-single-pole-physics

Comment on the quote: Because in my view I consider the electrons having one electrical charge and one of two magnetic charges, I think we have a nice equal footing of electricity and magnetism… (End of the comment.)

__________

Back to CERN and stuff. Last month it came out that the MoEDAL experiment has failed in the sense that no magnetic monopoles were observed. Here is a small screenshot from the preprint archive stuff:

Comment: No idea what these people are talking about when they talk about 68.5 times the electric charge… Are they talking about electric charge or magnetic charge?
(End of comment)

Source of the content of the picture above:

detector in 2.11 fb−1 of 13 TeV proton-proton collisions at the LHC.

https://arxiv.org/pdf/1712.09849.pdf

After a bit of searching I found back this beautiful video, coming from CERN, explaining how to find magnetic monopoles. It is clear they never ever studied the electron.

Yeah yeah my dear average CERN related human; what exactly is a magnetic monopole?

Does it have electric charge too and why should that be?

In my view where the electrons carry both electric and magnetic charge, a magnetic monopole with zero electric charge just does not exist.

__________

Ok, let me bring this post to an end by observing that at CERN they were not capable in the year 2017 of detecting the magnetic monopole as it should exist following the lines of thinking like Paul Dirac once did.

So that is a good thing because after thinking about four years about magnetism it would be horrible for me to find that at CERN they had a major discovery about magnetic monopoles…

Sorry CERN folks, your failure to find magnetic monopoles your way does not prove that electrons are indeed carrying magnetic charge. It just makes it a little bit more plausible that they do…

So my dear CERN folks, thanks for publishing your failure because for me it is another tiny quantum move into the direction of accepting the electron as it is.

__________

End of this post.

Intro to the calculation of the seven dimensional number tau (circular version).

All details will be in the next post but I succeeded into using matrix diagonalization in order to find this seven dimensional number tau.
For people who do not understand what a number tau is, this is always the logarithm of an imaginary unit. Think for example at the complex plane and her imaginary unit i. The number tau for the complex plane is log i = i pi/2.

The problem with finding numbers tau becomes increasingly difficult as the number of dimensions rise. I remember back in the year 2015 just staring at all those matrices popping up using internet applets like the next one:

Matrix logarithm calculator (it uses the de Pade approximation)
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

Yet back in the year 2015 I was riding on my noble iron horse (a cheap bicycle) through the swamps surrounding the village of Haren and suddenly I had a good idea. Coming home I tried the idea of matrix diagonalization out in 3 dimensions and it worked.

I even wrote a post about it on 23 Nov 2015:

Integral calculus done with matrix diagonalization.

Now I think that most readers who visit this website are familiar with the concept of finding a diagonal matrix D containing all eigenvalues of a given matrix M. Once you have the eigenvalues you can calculate the eigenvectors and as such craft your matrix C containing all eigenvectors.
You can write the stuff as next: D = C^-1 M C.

Suppose you don’t know what M is but I give you the matrices D, C and the inverse of C. Can you find the matrix M?

Yes that is a beerwalk, all you have to do is calculate M = C D C^-1 and you are good to go.

But with the logarithm comes a whole lot of subtle things for making the right choice for the eigenvalues that you place inside the diagonal matrix D. It turns out you only get the desired result if you use arguments in the complex plane between minus and plus pi.
This is caused by the fact that you always need to make a cut in the complex plane if you want to work with the complex logarithm; but it is a bit surprising that only the cut where you leave out all real negative numbers (and zero of course) makes the calculation go perfect and in all other cases it ends in utter and total disaster.

In the next three pictures I show you some screen shots with numerical values of matrix representations and the logarithm of those matrix representations.

The goal is to find mathematical expressions for the observed numerical values that are calculated via the above mentioned de Pade approximation. We don’t want only numerical approximations but also catch the stuff in a mathematical formulation.

At the end of the third picture you see the end result.

So it took some time to find this result, I wasted an entire week using the wrong cut in the complex plane. And that was stupid because I had forgotten my own idea when riding my noble iron horse through the Harener swamps…

The result for the seven dimensional number tau (circular version) as calculated in the next post is a blue print for any dimension although I will never write stuff down like in a general dimension setting because that is so boring to read.

Ok, see you around my dear reader.

On the derivative and integral of the inverse function.

After all that magnetism stuff it is about time to throw in a tiny bit of simple math around how to find the derivative and primitive (the integral) of the inverse of a function.

In most (introductory) textbooks on calculus you will find a nice way of finding the derivative of the inverse of a given function f(x) defined on the real line. For integration where you need to find the anti-derivative there is also a very elegant way of calculating those, but in my life I have never ever seen it in print on paper in an actual existing book.

Now last week I came across a video where another guy claimed that finding the primitive in this way was completely new but within 60 seconds with the help of the Google search engine you can find this is not the case.

According to a wiki on the subject of integration of the inverse of a function, the first know results date back to 1905. This is a remarkably short time ago and for myself speaking I think that many folks found this way too but for some strange reasons it never popped up to the surface. It is strange to observe that for example the method of the calculation of variations was invented included those fine differential equations that form the way to find for example the path of least action or minimal time but somehow those people never found the way to integrate the inverse of a function…

On the other hand, I have seen it myself that there can easily be a complete vacuum in mathematics; in my first year at the university I invented the so called product integral. Normally when you calculate an integral you can view that as adding up all the area under the graph of a certain function, with a product integral you do the same but you do not add it up but you multiply all stuff.
And in it’s most natural setting you do that with raising a function f(x) to the power dx.
That was my invention but although product integration has been studied for over a century, nobody had ever taken a function to the power dx…

Now enough of the blah blah blah done, this post is four pictures long and the wiki stating this cute formula was found in the year 1905 is the next:

Integral of inverse functions
https://en.wikipedia.org/wiki/Integral_of_inverse_functions

This post is four pictures (550 x 775 pixels), here they are:

So that was it for this post, see ya around my dear reader.

Updated on 16 Oct 2017:
Today I found that video back where some guy made those unsubstantial claims that this result was never ever found in the entire history of mathematics. That is not true but it is strange that the derivative is in every introductory course or book while the integral version is always absent.
We can safely jump to the conclusion that the integral version is not widespread known and this causes authors of those books not to include it.

The video goes under the title:
Rare Integration Strategy – You won’t learn this in Calculus.

So that was it for this update on this post, see ya around my dear reader.

Updated on 25 June 2018:
By sheer coincidence I came across a very nice video today. I remember that I wanted to discuss the situation as described into the video but I also want to keep the writing short at about 500 words.

So I skipped a discussion as where the function f(x) is hard to find but it is better to attack it via the inverse function. I know this sounds a bit vague but in the video you have such a situation.

The video goes under the title:
Integration Problem: Thinking Outside the “Box,” or the Given Region (From Stanford Math Tournament)

It seems to be from the year 2014 so it is refreshing to observe that it is not true that all math departments around the world are only occupied by zombies…

So that’s it for this update on this post.