What is the repeated conjugate ‘determinant style’? (Also: Repeated adjoint matrix.)

3D complex numbers, Matrix representation

This post is easy going for most people who have mastered the art of finding the inverse of a square matrix using the so called adjoint matrix. I was curious what happens to a 3D circular number if you take the conjugate ‘determinant style’ twice. In terms of standard linear algebra this is the same as taking the adjoint of the adjoint of a square matrix.

It is well known by now (I hope anyway) that 3D complex and circular numbers contain a set of numbers with a determinant of zero, you can’t find an inverse for them. To be precise, if you take some circular 3D number, say X, and you make some limit where you send X into a not invertible number, you know the inverse will blow up to infinity.
But the conjugate ‘determinant style’ does not blow up, on the contrary in the previous post we observed that taking this kind of conjugate gave an extra zero eigenvalue in this conjugate.

In terms of linear algebra: If a square matrix is not invertible, it’s adjoint is ‘even more’ non invertible because a lot of the eigenvalues of that matrix turn to zero.

And although the inverse blows up to infinity, the cute result found is that it blows up in a very specific direction. After all it is the fact that the determinant goes to zero that blows the whole thing up, the conjugate ‘determinant style’ is as continuous as can be around zero…

It’s a miracle but the math is not that hard this time.

Four pictures for now and I plan on a small one picture addendum later.
So lets go:

Isn’t it cute? This infinity has a direction namely the number alpha…;)
Small correction: It should be taking the conjugate TWICE…

All in all this is not a deep math post but it was fun to look at anyway. May be a small appendix will be added later, so may be till updates inside this post or otherwise in some new post.
Added Sunday 16 April: A small appendix where you can see what the adjoint taking process is doing with the eigenvalues of a 5×5 diagonal matrix. The appendix was just over one picture long so I had to spread it out over two pictures. You understand fast what the point is if you calculate a few of the determinants of those minor matrices. Remark here with a 5×5 matrix all such minors are 4×4 matrices so it is the standard setting and not like that advanced theorem of Pythagoras stuff.
Well it all speaks for itself:

Ok, that was it for this update. Thanks for the attention and see you in another post.

Factorization of the determinant inside the space of 3D circular numbers. Aka: The conjugate ‘determinant style’.

3D complex numbers, Matrix representation

A few weeks back I was thinking in writing finally some post about general theory for spaces with arbitrary dimension. It soon dawned on me that the first post should be about the impossibility of solving X^2 = -1 on spaces of odd dimension for both the complex and the circular method of mulitplication on those spaces. So post number one should be about the fact the famous number i does not exist in spaces with dimensions 1, 3, 5 etc.
And what about the second post? Well you can always factorize the determinant inside such spaces, that is a very interesting observation because the determinant is also the product of all eigenvalues. These eigenvalues live traditionally in the complex plane and as such a naive math professor could easily think that the determinant can only be factorized inside the complex plane. So that would be a reasonable post number two.
Since all these years I only did such a factorization once I decided to do it again and that is this post. The basic idea is very simple: If you want to find an expression for the inverse of a general 3D circular number, you need the determinant of that number. From that you can easily find a factorization of the determinant. It’s as simple as efficient.

But now I have repeated it in the space of 3D circular numbers I discovered that part of this factorization behaves very interesting when you restrict yourself to the subset of all 3D circular number that are not invertible. That is that taking the conjugate ‘determinant style’. The weird result is that taking this kind of a conjugate increases the number of eigenvalues that are zero. So this form of conjugation transports circular numbers with only one eigenvalue zero to the sub-space of numbers with two eigenvalues zero.

For years I have been avoiding writing general theory because I considered it better to take one space at a time and look at the details on just that one space. May be that still is the best way to go because now I have this new transporting detail for only what would be the second post of a general theory, it looks like it is very hard to prove such a thing in a general setting.

Luckily the math content of this post is not deep in the sense if you know how to find the inverse of a square matrix, you understand fast what is going on at the surface. But what happens at the level of non-invertibles is mind blowing: What the hell is going on there and is it possible to catch that into some form of general theory?

I tried to keep it short but all in all it grew to a nice patch of math that is 8 pictures long. Here is the stuff:

At the end of this post I want to remark that the quadratic behaviour of our conjugate ‘determinant style’ is caused by the fact it was done on a 3D space. If for example you are looking at 17 dimensional number, complex or circular, this method of taking a conjugate is a 16 degree beast in 17 variables. how to prove all non-invertible numbers get transported to more and more eigenvalues zero?

May be it is better to skip the whole idea of crafting a general theory once more and only look at the beautiful specifics of the individual spaces under consideration.

End of this post and thanks for your attention.

Comparison of the ‘Speed = the Square’ equation on 7 different spaces.

3D complex numbers, 4D complex numbers, Uncategorized

This post is very simlilar to a few back when we calculated the results on 4 different spaces. This time I hardly pen down any calculation but only give the results so we can compare them a little bit.
The way most professional math professors tell the story of complex numbers it goes a bit like this: We have the real number line, the complex plane and on top of that a genius named Hamilton found the quaternions. On top of that there are a bunch of so called Clifford algebra’s and oh we math professors are just so good. There is no comparison to us, we are the smartest professionals in the world!

Well that is very interesting because it is well known these so called ‘professionals’ could not find the 3D complex numbers for about 150 years. So how come they all say we have this and that (complex plane and quaternions) and that’s enough, we are just perfect! Why they keep on saying rubbish like that is the so called Dunning-Kruger effect. That’s something from psychology and it says that people who lack understanding of some complicated stuff also lack the insight that they are stupid to the bone when it comes to that particular complicated stuff. So the views of professional math professors is very interesting but can be neglected one 100 percent, it’s just Dunning-Kruger effect…

If you look at the seven results of the ‘Speed = the Square’ equations, the solutions form a strickt pattern that only depends of the number of dimensions and if it is the complex or the circular multiplication. So every time a math professor goes from the complex plane to the wonderful world of quaternions you now know you are listening to a weirdo.

I said I only give results but since I have never ever introduced the 4D circular numbers I just extrapolated the other six spaces to the solution that lives in that beautiful space. So the last example is a bit longer.

Anyway although the math depth of this post is not that very deep (solving a differential equation that wants the derivative to be the square of what you differentiate), it clearly demonstrates solutions of all 7 different spaces look strikingly similar.
But because of the Dunning-Kruger effect likely the math professors will keep on telling total crap when it comes to complex numbers. Why am I wasting my time on explaining math professor behaviour? Better go to the five pictures of our post. Here we go & bye bye math professors.

May be I should write some posts about general complex number theory on spaces of arbitrary dimension. On the other hand I found the 3D complex numbers back in the year 1990. So if after all those years I will once more try to write some general theory one thing will be clear: Math professors will keep on trying to convince you of the beauty of quaternions or that garbage from the Clifford algebra’s.

Why, as a society, do we keep on wating tax payer money on math professors? Ok, they do not everything wrong but all in all it is not a great science or so where the participants are capable of weeding the faults out and grow more of the good stuff.
Let me end this post and thank you for your attention.

Correction: My spinning plasma model for sun spots is likely not correct.

Corrections

Oops I likely made a many year mistake when it comes to the magnetic stuff. Many years ago I had the idea that the magnetism as found in sun spots could possibly explained by spinning plasma underneath the solar surface.
After all if electrons are magnetic monopoles, a spinning cylinder shaped plasma should eject lots of electrons along it’s magnetic field lines. That makes the spinning plasma a terrible good magnet because is a lot of positive charged plasma is spinning that creates strong magnetic fields.

After the original idea it took me about one year there could be a possible mechanism on the sun that creates such spinning plasma structures: The sun rotates faster at the equator as it does at the poles.

Now sun spots often come in pairs with opposite magnetic polarity and in my view I thought the leading sun spot was the one created by a bunch of rotating plasma under it.

It is easy to understand that if the root couse of sun spots was a rotating column of plasma underneath them, on the opposite hemispheres of the sun the leading sun spot should have opposite magnetic polarity. That one always checked true, but for the rotational hypothesis to be true over the solar cycles the polar magneticity of the leading sun spot should always be the same.

And that is where likely my old idea is crashing right now. In the next picture you see what is more or less observed in the last change of the solar cycle, for me this is not funny.

SC24 and SC25 stand for Solar Cycle 24 & 25 and again: for me this is not funny:

Yes it is what it is. But at least as soon as I discover I have made a serious mistake I tell that as soon as possible.
All in all this mistake does not have any impact on the tiny fact that it is impossible for electrons to be tiny magnets, electrons are magnetic monopoles and as such we have two variants of them. So the Gauss law for magnetism is just not true for an individual electron, it is nonsense to say magnetic field lines always loop in on themselves.

But after seven years of explaining this kind of mistakes, that stuff known as the science of physics is not capable of cleaning herself of stupid ideas.

Let’s leave it with that, this correction is a set back but the weirdo’s classified as the physics professors still have to give some experimental proof that electrons are indeed ‘tiny magnets’.

Solving the ‘Speed = the Square’ equation on the space of 4D complex numbers.

3D complex numbers, 4D complex numbers

Unavoidable I had to write some post after the video on the quaternion from Hamilton. Now my 4D complex numbers commute so they are very different from the standard version of quaternions. Just like in the complex plane the multiplication is ruled by the imaginary unit i that has the defining property of i^2 = -1. On the space of four dimensional complex number I mostly write l for the first imaginary component, the defining property is of course that now the fourth power equals minus one: l^4 = -1.
In 2018 I wrote about 20 introdutionary posts about the 4D complex numbers. That is much more as you would need for the quaternions of Hamilton but on the quaternions you can’t do complex analysis and that explains almost all of the difference.
You can view the quaternions as three complex planes fused together by the common use of the real line. My 4D complex numbers can be viewed as a merge of two complex planes in the sense that there are two planes clearly ‘the same’ as a complex plane.
This post is once more one of the ‘Speed = the Square’ equations and just as on the other spaces we looked at we choose the initial condition such that it is the first imaginary unit l. As such our solution is easily found to be f(t) = l / (1 – lt) because if you differentiate that you get the square. So from the mathematical point of view this is all rather shallow math because all we have to do is find the four coordinate functions of our solution f(t). For that you need to calculate the inverse of 1 – lt and to be honest after so much years I think almost all math professors are just to fucking stupid to find the inverse of any non real 4D compex number Z let alone if you have something with a variable t in it like in 1/(1 – lt).

I did my best to write this as transparant as possible while also keeping it as short as possible. For an indepth look at how to find the inverse of a 4D complex number, look for Part 17 in the intro series to the 4D complex numbers. (Just use the search function for this website for that.)

This post is just three pictures long so lets hope that is inside your avarage attention span. And it’s math so without doubt a lot of people will digest this with a speed of one picture a week! No I am not being sarcastic or so, I just like as how I evolved to the math place I am now. Often that also goes very slow but it has to be remarked the math professors are much more slow slow slow because they could not find the 3D complex numbers in all of human history.
Let’s dive into the picture stuff:

One of the funny things of the math of this post is that on the one hand it is very simple: You only need high school math like the quotient rule for checking my claims are true and differentiation mimics the multiplication on the 4D complex numbers. On the other hand you have those math professors likely not capable of finding these easy coordinate functions for themselves.
But this post is not meant as an anti math professor rant but more upon the beauty of simple math you can do on say the space of 4D complex numbers.
See you in the next post.

Why could Hamilton not find the three dimensional complex numbers?

3D complex numbers, 4D complex numbers

This very short post was written because of a video from the video channel Kathy loves physics. It is one of those “Quaternions are fantastic” video’s. And Kathy just like a lot of other physics people think indeed that quaternions are fantastic. But you cannot differentiate or integrate on the quaternions so I guess this stronly limits it’s use in physics.
But quaternions are very handy in describing rotations in 3D space, I never studied the details but it was said that on the space shuttle it was used for nagvigation. And because of these rotation properties at present day they are used in the games industry.

Anyway in the video Kathy explains that Hamilton did try for a long time to find the three dimensional complex numbers. And he never succeeded in that. Of course I know this for decades right now but in the past I never looked into a tiny bit more detail in what Hamilton was actually doing.
And he was looking at complex numbers of the form X = x + yi + zj where the imaginary components both equal to minus one: i^2 = j^2 = -1.

If you check the easy calculations in this post for yourself, it is amazing how much it already looks like the stuff as found on the quaternions. As such it is all of a sudden much less a surprise that Hamilton found the quaternions. As a matter of fact it was only waiting until he would stumble across them. But at the time the concept of a four dimensional space was something that made you look like a crazy lunatic, there were even vector wars and lots of crazy emotional stuff.

At present day it is accepted that 3D complex numbers do not exist, in my experience the professional math community is still emotionally laden but now into the direction of total neglect. Stupid shallow thought like “If Hamilton could not find them, they likely don’t exist”.

Back in the 19-th century they were always looking for an extenstion of the complex plane to three dimensional space. Of course they failed in that attempt because it is a fact of math life that you cannot solve the equation X^2 = -1 on the space of 3D complex (and also circular) numbers.

The content of this post is just two pictures, after that two more pictures as I used them on the other website and after that you can finally dive into the Video from Kathy. If you are interested in physics and also the history of physics, Kathy her channel is a thing you should take a look at if you’ve never seen it. Here we go:

YES, that is what he should have done. Hamilton tried for about one decade to find the numbers that form the title of this very website, so may be he tried this kind of approach. I don’t know, but the 3D complex numbers are not some extension of the complex plane because 2 is not a divisor of 3. You know that prime number stuff is going on here.
But the math professors are not interested in that kind of stuff.

Here is how I used it on the other website:

As you see in the above picture I was already working on the previous post because if you differentiate the three functions that mimics the 3D circular multiplication. You can also mimic the multiplication on the complex plane, that is in the next picture:

At last you can view the famous video of Kathy! It’s only 30 minutes or so but if you see too many so called TIKTOK videos that is infinitely long: Wow 30 minutes long looking at just one video?

End of this post, likely the next post is about 4D complex numbers.

Solving the ‘Speed = The Square’ equation on four different spaces.

3D complex numbers, Matrix representation, Uncategorized

With ‘speed = square’ I simply mean that the speed is a vector made up of the square of where you are. The four spaces are:
1) The real line,
2) The complex plane (2D complex numbers),
3) The 3D circular numbers and
4) The 3D complex numbers.

I will write the solutions always as dependend on time, so on the real line a solution is written as x(t), on the complex plane as z(t) and on both 3D number spaces as X(t). And because it looks rather compact I also use the Newtonian dot notation for the derivative with respect to time. It has to be remarked that Newton often used this notation for natural objects with some kind of speed (didn’t he name it flux or so?).
Anyway this post has nothing to do with physics, here we just perform an interesting mathematical ecercise: We look at what happens when points always have a speed that is the square of their position.

On every space I give only one solution, that is a curve with a specific initital value, mostly the first imaginary component on that space. Of course on the real line the initial condition must be a real number because it lacks imaginary stuff.

If you go through the seven pictures of this post, ask in the back of your mind question as why is this all working? Well that is because the time domains we are using are made of real numbers and, that is important, the real line is also a part of the complex and circular number systems.
The other way you can argue that the geometric series stuff we use can also be extended from the real line to the three other spaces. To be precise: we don’t use the geometric series but the fractional function that represents it.

Ok, lets go to the seven pictures:

That Newton dot notation just looks so cute…
The words ‘Analytic continuation’ are not completely correct…

Remark: This post is not deep mathematics or so. We start every time with a function we know that if you differentiate it you will get the square. After that we look at it’s coordinate functions and shout in bewilderment: Wow that gives the square, it is a God given miracle!

No these are not God given miracles but I did an internet search on the next phrase of Latex code: \dot{z} = z^2. To my surprise nothing of interest popped up in the Google search results. So I wonder if this is just one more case of low hanging math fruits that are not plucked by math professors? Who knows?

End of this post, thanks for your attention.

A simple example showing the invariance of the determinant (so it returns always a positive number).

Pythagoras stuff

This is one of the details I should have posted last year. So this post is some mustard after the meal. The content is just two pictures long. In it I show you how to calculate the area of a parallelogram in 4D space. After that we swap the two columns and use the same method again. In both cases the area of the parallelogram equal the square root of 500.
If you read stuff from this website you likely have enjoyed some classes in linear algebra, likely you know that if you swap two columns (or two rows) in a square matrix, the determinant changes sign.
But the way we turn a non-square matrix into a square matrix is done in such a way that it has to return a positive (or better: non-negative) number.
In this example you can see that if you swap the two spanning columns of the parallelogram, the first extra column or the third column in our final matrix also chages sign. So the overall determinant of the 4×4 final matrix ‘observes’ a swap in the first two columns and also a swap in the sign of the third column. Hence the determinant does not change sign…

Originally I only needed a few cute looking formulas for use on the other website. That are the two matrices below. But when finished I added some text and as such we have a brand new post for this website.

In this example I did not normalize the extra columns to one so if you want you can play a bit with it and as such observe how their norms are related to the area of the diverse parallograms in here. For example if you calculate the norm of the fourth column, it is the square root of 75,000 while the determinant of the whole 4×4 matrix is 75,000.

As such constructing square matrices like this always leads to the last column having a norm that is the square root of the determinant.
That is a funny property, or not?

Anyway here are the two pictures, the third picture is an illustration of how it was used on the other website. As usual all pictures have sizes of 550×825.

In the third picture I used an old photo of Brigitte Bardot as a background picture. Now both Brigitte and me we looked a lot more fresh back in the time from before they invented the stone age. Our minds were sharp and our bodies fast while at present day we are just another old sack of skin filled with bones, fat and some muscle. Life is cruel..

Ok, lets end this post now and see you around my dear reader.

Example: How to turn a 4×1 column into a 4×4 square matrix.

Pythagoras stuff

Yes I know that two posts back I said that this would be the last time we would do Pythagoras stuff like this. On the other hand I was very unsatisfied with that post (title: That weird root formula). Also I had wanted to post the math below before but I did not have the time.
All in all since I was horribly bad in the post upon that weird root formula about how to make extra columns, may this short post compensates a bit for that.
This post starts with a column of four real numbers, say a, b, c and d. The goal is to keep on adding extra columns such that all columns are perpendicular to each other. Don’t confuse that with an orthogonal matrix, orthogonal matrices also have all their columns perpendicular but the columns are also all of norm one.
For myself I name matrices that have all their columns perpendicular to each other ‘perpendicular matrices’ but this is not a common thing in math communications as far as I know.
I show you two examples here: First I make an extra column based on the a and b entry of our vector. In the second expansion of the same vector I use the middle two entries b and c.
This should serve as examples that make it as transparant as possible how you must use the +/- chessboard pattern that comes with calculations like this.
For understanding this post it comes in very handy if you have done & understand the general way of crafting the inverse of a square matrix. I think most people will see the brilliant +/- chessboard scheme there for the first time in their lives.

I don’t know much about the history of math, but I like it that the +/- chessboard scheme has no human name attached to it like in “Hilbert space”. I guess this chessboard pattern emerged slowly over the cause of a few decades with contributions of many people. So in the end there was nobody to name it too because this big success just had to many fathers.

Another explanation for the lack of a human name to the famous +/- chessboard pattern is that the person who for the first time chrystal clear wrote out the stuff, this person was not an overpaid professional math professors. But say an amateur just like me. Well in those good old times just like now, the overpaid math professors can’t give credit to such an undesireable person of course…
Yet not all is negative when it comes to professional math professors: They are still very good at telling anybody who wants to hear it that: “We tried but we could not find the three dimensional complex numbers”.

After all that human blah blah blah, why not take a look at the three pictures?

Please do that exercise so you can say you understand that +/- chessboard scheme.

That was it for this post.

Two more videos that explain electron spin wrong.

Magnetism, Uncategorized

A happy new year by the way, it is now 3 Jan over here so it is not too late to wish you that. So be happy if you can ask a physics professor or teacher as why there is no experimental proof at all that electrons are tiny magnets. And if the answer is not satisfactory, just chop the head of while being happy…;)

But serious, I selected the first video because the guy from the Science Asylum channel gives are very tiny estimated upper bound for the possible size of the electron: 10 to the -18 power meter as diameter.
That is very very small, it is a nano nano meter.

Lets construct a so called ‘toy model’ for imitating in a simple manner how the electron is supposed to be a tiny magnet: Take two pointsize magnetic monopoles, a north and a south one and place them 10 to the power -18 meter apart. Lets name this distance d.
An important feature of such a dipole is that it’s magnetic field declines inversely with the third power of d.

Let me give you an example: Take a line through the north and south pole of our toy electron and go out a distance of say 10d above the north pole. So the distance of our point on that line is 10d to the north pole and 11d to the south pole. The magnetic forces or field strength if you want is now proportional to 1/10^2 and 1/11^2. But north and south pole have opposite workings so we are looking at the difference: 1/10^2 – 1/11^2 and that is something of the order 1/1000.

If the electron diameter is indeed at most this distance d, in that case the two overlapping magnetic fields cancel each other almost out. If all that tiny magnet stuff is true, in that case the electron should be magnetically neutral. In a constant magnetic field that does not vary in space, by definition this tiny magnet electron should be neutral (if it all was true).

Let me show you two screen shots from the video from the Science Asylum. The first simple shows you the claim the electron has at most this size d.

On a nano nano scale this should be magnetically neutral…

A long time ago I estimated the result in next picture too but I always used an electron diameter of 10 to the -16 power, so one hundred time as big as the Asylum guys claims. Anyway there is nothing spinning over there because it must rotate a huge multiple of the speed of light. Now we can honestly say that Albert Einstein did not understand much about electron spin, but we can safely conclude that electron spin is not related to rotation of a spherical charged body the size of d.

One million times the speed of light…

Ok, let me hang in the video where we have once more the implicit claim that magnetism is always a magnetic dipole without one iota of experimental proof for that claim:

In my view the most misleading name is spin, it sets your brain totally wrong.

In the next video you see a guy at work showing that the oxygen in the air you breathe is magnetic. The magnetic properties of oxygen are truly breathtaking because it has to do with a so called ‘non-binding’ electron pair. In chemistry a non-binding electron pair is a pair with the same electron spin. Weirdly enough the physics professors keep their mouth shut: All electron pairs obey the Pauli exclusion principle!
Until it doesn’t like in molecular oxygen.

But I digress, the reason I selected this video can be found at 3.40 minutes into it: The guy ‘explains’ the behavior of the oxygen by stating that the two electrons in the non-binding pair align their magnetic dipole to the applied magnetic field. The problem with this kind of ‘explanation’ is that it does not explain as why the electrons get accelerated. As said above; if electrons are tiny magnetic dipoles, they are basically magnetically neutral. And we are to believe that the oxygen molecules get accelerated by the applied magnetic field because two little electrons ‘align their dipole magnetic moment’. Give me a break: that is crap and the next stuff look much more logical and observable:
Electrons are not magnetic dipoles but magnetic monopoles.

Here is the second video:

The reason for posting this second video is that I often obverve people from physics thinking that the alignment or for that matter the anti-alignment explains the acceleration and forces involved.

After seven years into this stuff I only wonder:

Why do the physics professionals like teachers and professors not see they are telling utter crap? Why are they so fucking stupid all of the time?
End of this post. Once more: A happy new year.