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.
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.
There is a Youtube channel named the science asylum. It is run by a guy that makes your head tired but for a few minutes it is often nice to watch. In the next video he tries to explain magnetism, both electro-magnetism and permanent magnets.
Here is the video:
The video guy tries to explain permanent magnets via the concept of angular momentum. As far as I know physics now this is indeed the way it has historically evolved. Now a lot of physics folks think that the unpaired electrons somehow go around the nucleus and as such create a magnetic dipole moment, other place more emphasis on electrons being the basic dipole magnets and if they align all conditions for possible permanent magnets are there. Of course beside electrons also the so called magnetic domains need to align and the official version is that this is more or less the way some materials like iron can be turned into permanent magnets.
In my view there is a long list of problems that come with that. For example the Curie temperature, above the Curie temperature the permanent magnet looses it’s magnetism completely.
If you view electrons as magnetic monopoles, stuff like the Curie temperature become a lot more logical: above the Curie temperature it has to be that the metal is so hot it starts to loose it’s unpaired electrons.
In my view, when permanent magnets are manufactured it is the applied external magnetic field and the slow slow cooling down from above the Curie temperature that ensures the magnet becomes ‘permanent’.
There should be a simple experimental proof for that but although for years I tried to find it, until now I still haven’t found it. But in our global industrie a lof of permanent magnets are made every day and the experimental proof for my view on electron spin simply says the next:
If during the cooling down a NS external magnetic field is applied, the permanent magnet will come out as a SN magnet.
With SN I simply mean that the magnetic south pole is on the left while the magnetic north pole is on the right. So the macroscopic object known as the permanent magnet will always be anti-aligned with the magnetic field that is applied during the fabrication of the permanent magnet.
I still haven’t found it after all these years, but if it comes out that permanent magnets have the same orientation as the applied external magnetic field during fabrication, I can trash my theory of magnetic charges and finally go on with my life…
But there are plenty of problems more with the standard model version of electrons being magnetic dipoles. Here is a screen shot from 4.35 minutes into the video from the crazy asylum guy and it contains so called quantum numbers.
The title in the picture might look a bit confusing because the only single-electron atom in the universe is the hydrogen atom. Likely the author means the number of unpaired electrons in some atom.
You observe two quantum numbers that are holy inside quantum mechanics:
The so called ‘Total orbital angular momentum’ number l and something weird because it is only along the z-axis direction m_l.
In my view, if electrons carry magnetic charge and are not magnetic dipoles and for example it has 3 unpaired electrons, total magnetic charge runs from -3 to +3.
And that is precisely what the m subscript l number does…
So I still have to find the very first fault in my simple idea that electrons just carry one of the two possible magnetic charges.
In this post I skipped the fourth quantum number: total spin. That is also a quantum number thing so is it a vector (a magnetic dipole) or a magnetic charge (a magnetic monopole)?
Ok, let’s leave it with that. Have a nice set of quantum numbers or try to get one.
Added on 26 Nov:
May be looking at chaotic guys makes me chaotic too, but reading back what I wrote yesterday I think it is better to explain the ‘anti alignment’ a bit more because I explained it far too confusing I just guess.
I made a schematic sketch of it, you see two coils that make a long lasting constant magnetic field and in that field a hot piece of iron is slowly cooling down. An important feature of iron is that the four unpaired electrons are in the inner shells, this is a consequence of the so called aufbau principle.
Here is the sketch:
So my basic idea of manufacturing permanent magnets is the fact that during the long cool down the unpaired electrons can settle in those inner shells inside the electron could of the iron atoms.
Furthermore it has to be that the chrystal structure metals like iron make is such that the individual iron atoms cannot rotate. It is fixed in place. And with the unpaired magnetic charge carrying electrons in place after the long cool down, that is why your permanent magnet is more or less permanent:
Small surplusses of south charged electrons should be at the left & vice versa a bit more north magnetic charge at the right as sketched above.
This all sounds very simple but we also have those magnetic domains in metals like iron, those magnetic domains are just small surplusses of either one of the two magnetic charges. This blurs the simplicity a bit but it perfectly explains as why unmagnetized iron gets attracted to magnets anyway: the dynamics of the magnetic domain changes under the influence of the outside permanent magnet…
At last I want to remark that if the idea of permanent magnetism would solely based on dipole electron spin that aligns, in that case strong permanent magnets should change the permanent magnetism of weakly magnetized iron. The strong magnets would simply eat up the electrons of the weak permanent magnet.
That just does not happen. Last spring I made a small toungue in cheek ‘experiment’ with that where I placed my stack of most strong neodymium magnets against two of my most weak magnets. It was all fixed in place over 24 hours, but after all that time my two most weak permanent magnets exposed the same behaviour as before.
So no electron was flipped and for me that was one more reason to say farewell to the idea of electrons being magnetic monopoles.
Here is a link to that very simple toungue in cheek experiment, I hope it is so simple that physics professors will vomit on that…
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…)
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.
This post is on magnetism only but it is more or less in the planning that the next post on 4D complex numbers is about diagonalization of a 4D complex number. After that we have (may be) to try and calculate the number tau in this way.
Ok magnetic stuff:
Since I think that electrons are not magnetic dipoles but carry one of the two possible magnetic charges, electrons will always be accelerated by magnetic fields just as they are by electrical fields.
That basic idea of magnetic charge makes a whole lot of things much more easy to understand. For example electron pairs are only observed as pairs, never as triples or whatever what like circular structures of 17 electrons…
Let’s look at the most simple molecule we know of: molecular hydrogen. Two atomic hydrogen are both electrically neutral, why should they bond anyway? But if the two electrons of those two hydrogen atoms carry opposite magnetic charge, that could be a reason as why they want to bond anyway.
And if you think about it that way (the protons also carry some magnetic charge) all of a sudden a hydrogen molecule is nothing but a balancing act between electric and magnetic charges.
Once more: Why do we only observe electron pairs in chemical bonding via electron pairs? If the electron was indeed a magnetic dipole, what explains we only observe electron pairs?
I made a beautiful drawing of two electron magnetic dipoles, I know I know it is a bit simplistic with two simple lobes of magnetism in a 2D representation. But here it is & how is all that bonding in the electron pair supposed to go?
Why do we only observe electron pairs and not other forms of possible magnetic dipole formations? We have never seen only one electron triple as far as I know.
I know it is a terrible drawing. Always when I show drawings to other human beings they always get tears in their eyes…:)
In the beginning I just thought that the tears were explained because those people could see the underlying beauty of my drawings. But later they also started vomiting and I was just scratching my head and decided to post less and less drawings because that vomit is so smelly and dirty.
But let’s not talk about the smell of vomit but a bit more on the wonderful progression the Max Planck Institute is making with their expensive nuclear fusion reactor known as the Wendelstein 7-X fusion reactor. They have set breathtaking records like in the next link:
If my version of electron spin carries more truth compared to the retarded version of every electron is a magnetic dipole, if that is true nuclear fusion machines like Tokamak or the Stellarator design as used in the Wendelstein 7-X reactor will not work properly.
So let’s make it a little contest between me and the entire Max Planck Institute, in particular the IPP parts of it (IPP = Institute for Plasma Physics).
Now that is German stuff so I will use a German video as found on the internet where a German guy named Harold Lesch asks questions at another German guy named Hartmut Zohm. One of those questions is answered at 3.33 minutes into the video where Harold asks if there are still some major ‘technical hurdles’ that have to be taken…
Hartmut Zohm says: Das haben wir alles schon ausgeraumt.
Loosely translated the answer says that ‘everything has been cleaned out’.
Here is the video:
Harald Lesch & Hartmut Zohm zur Fusionsforschung
And of course a picture of how it looks to observe two humans not understanding electron acceleration by magnetic fields:
So what is the little contest between the Max Planck Institute and me?
Very simple: They will keep on going with the insight from the standard model of physics that says electrons are magnetic dipoles. And from my side I will keep on insisting that electrons are the long sought magnetic monopoles and as such are accelerated by magnetic fields.
It will take some time to let this pan out, after all the Wendelstein 7-X stellarator is not something you turn on like a light bulb. But one thing is clear; the longer the machine is turned on the longer the electrons get accelerated and likely this is creating a long list of plasma stability problems like ever growing turbulence. Or more and more electrons smashing into the vessel walls because the high speed will make them leave the magnetic field lines they follow at low speeds…
At last I want to remark that electrons as magnetic dipoles with only one of the two possible magnetic charges makes something like the formation of H2 easier to understand. But that comes with a hefty price: Now the formation of proteins becomes much more difficult to understand. If you look at animations from how proteins form very often you see the electron as a magnetic dipole and when needed it simply flips around… So what happens in reality is unknown to me, is there all kinds of electron transport going on so that the right electron is at the right place in a chemical reaction? Or are there all kinds of photons going round changing the magnetic charge of the individual electrons?
I don’t have a clue as how our human bodies actually make all those proteins…
That is the end of this post.
A correction on the 17 June 2017 post is added on 29 Oct 2018. It is just a silly typo in just an example of a possible so called trapdoor function, but last week when I read that old post again I decided to correct it. So although it is a very old post compared to the present day news cycle where idiots run the atheneum formerly known as the Federal government of the USA, I still live in timescales that evolve more slowly.
After all, almost all of the math as written down in the post about finding prime factors of huge composite numbers as they are used in internet security was found by me in the nineties. So it is over two decades old math and by other standards that is still very young math…
The original post was 15 pictures long, as usual it is very hard to understand all stuff in a single day. The human brain learns slow because all that brain matter has to rewire itself and that is a process that makes you tired, just like nature wants I just guess.
In the end I came close, I had all the technical stuff figured out but I still lacked a mechanism of converging into the direction of the prime numbers as used in for example communications encryption.
Ok, the correction itself is on the second picture of the old 17 June 2017 post.
If you don’t want to read the old post, the correction looks like this:
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.
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:
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.
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.
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:
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.
