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 supersized electron, can you accelerate one cubic cm of this stuff with a magnetic field?

This post is a continuation of Reason number 50 as why electrons cannot be magnetic dipoles as found on the other website. I published nr 50 in 2017 on 14 Oct. In that nr 50 Reason I tried to estimate the gradient of an inhomogeneous (non constant) magnetic field. It was just a rough estimation so you can have all kinds of critisism on it, but the gradient needed in the magnetic field was so huge that we safely can conclude that electrons cannot be accelerated by non constant magnetic fields.

Back in the year 2017 I more or less stated that universities are never very helpful. I joked that the word cooporation was not found in their dictionary. So now about 18 months later this seems to be true, why is that? Well all universities are relatively formal structures, most things go along some kind of protocol. For example when I would try to get a research job for the study of magnetic domains (because I think magnetic domains have surplusses of either one of the spin variants, so every magnetic domain is a magnetic ‘monopole’ on the domain level), that likely would not be possible. Because everything goes in such a formal manner likely I have to start as a first year student of physics, slowly climb the ranks and that’s it because ‘we cannot make an exception’. And from the university this is rather logical; if they give in to one weirdo that think that electrons cannot be magnetic dipoles, next comes along another crazy person that wants to study more homeopathic medicines or whatever what.

Ok, what are we looking at in this post? I simply view the electron as some kind of massive small sphere with a diameter of 10^-16 meter and as such estimate the density or the mass per cubic meter.

Without trying any kind of calculation, try to accelerate such an object with an inhomogeneous magnetic field…

It is three pictures long so this is a short post although the numbers are impressive. Picture sizes all 550×775 pixels.

Ok, this is what I had to say on the impossibility of accelerating magnetic dipole electrons in any meaningful amount while using inhomogeneous magnetic fields… See you in the next post.

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.

What would a quantum measurement on Paul his IQ yield?

Today I came across one of those video’s where people try to explain how permanent magnets work. And originally I thought of a title like ‘Idiot of the day observed’ but soon I changed my mind because Paul Sutter does not do it on purpose; what he says is more or less the general accepted version of permanent magnetism…

In general there are two lines of reasoning when it comes to permanent magnets: One line of reasoning is that the magnetic domains get aligned, the other way is that the electron spin of all unpaired electrons align.

Paul Sutter goes for the second line of reasoning; the spin of all unpaired electrons align giving rise to a permanent magnet. Just like everything else in the video it is just wrong; in my view where the electrons carry magnetic charge, it is the placement of the unpaired electrons in the inner shells of for example an iron atom that makes the global permanent magnet emerge.

If it was just electrons having all their dipole magnetic moments point in the same direction, in that case with a strong magnet you could always change or invert the magnetic direction of a weak magnet. In practice this just does not happen; last spring I even made a simple experiment with this: I took my stack of the most strong magnets I have and placed them over 24 hours against the two most weak magnets I have. And, like expected, there was no change at all in the weak magnets indicating there is some kind of threshold at work. The threshold is of course that it is hard to remove the magnetically charged electrons from the inner shells of the iron atoms…

Here are two pictures of the simple experiment from 08 March 2018; the permanent but very weak magnets on the left were exposed to the stack of neodymium magnets for just over 24 hours and just nothing changed in the behavior of the weak magnets. If electron magnetic moment alignment were a significant factor in permanent magnetism, the stronger permanent magnets should alter the magnetic properties of the weak magnets. It just does not happen…

Weak at the left, strong at the right.
After 24+ hours of waiting zero change observed in the weak magnets.

A link to what I wrote one year back on this very simple experiment is:
08 March 2018: Reason 56: This experiment shows zero spin torque transfer. http://kinkytshirts.nl/rootdirectory/just_some_math/monopole_magnetic_stuff03.htm#08March2018

It is lovely to see so many of the wrong stuff bound together in just one short video: For example when Paul explains why electron pairs are magnetically neutral while the unpaired electrons are not. In my view if it is true that electrons are magnetic dipoles, they would be magnetically neutral. They are not and in my view this shows electrons are magnetic monopoles. How does Paul explain it? Very simple: The electron pair is magnetically neutral because one of the electrons has spin up while the other has spin down and that cancels each other out.

Here is the Youtube video of Paul Sutter, please don’t think that Paul is a dumb person or so. This is just the view of professional physics folks that have studied magnetism for centuries…

Link: https://www.youtube.com/watch?v=6uwjqy2HCgY

May be at last I am getting a little bit sarcastic: They have studied that for centuries… But if you let go that dumb and unproven Gauss law for magnetism, just try to think about our physical reality as electrons also carry magnetic charge beside their electric charge, a lot of things become better to understand. After all, why do we only observe electron pairs? Why never something else like an electron triplet?

Ok, let’s leave it with that. Till updates my dear reader.

On Schrödingers cat & an example known as the envelope problem.

Today the Youtube channel SciShow had one more video out on quantum mechanics and as such the famous cat of the Schrödinger cat in a box problem comes along once more.

As usual we are told the cat can be in a super position of being alive and dead at the same time. I wonder why people think that this can be true, as far as I know history the Schrödinger guy came up with this example as an antidote as being everything into a super position…

I suppose you already know what the cat in the box setup is. The cat dies if just one radioactive atom decays yes or no. If you are outside of the box it makes sense to use a probalistic model of the situation, but does this mean that in reality inside the box the cat is dead and alive at the same time? After all the cat will be the very first to observe if radio active decay has happened because as soon as it does the state of the cat goes from alive to dead. So inside the box there is at least one observer present and as such all quantum states we are interested in (radio active decay yes or no) is constantly measured all of the time.

For myself speaking I use the fact that a cat cannot be in a super position of being alive and dead as an example that an individual atom cannot be in a state where radio active decay has passed yes or no.

That does not mean quantum particles cannot be in super positions, for example photons behave often like they took all possible paths to arrive somewhere. But as soon as there are all kinds of different energy levels involved this becomes more and more problematic. For example can a particle be in a super position of being a neutron and a proton? Can a particle be in a super position of being an electron and a positron? Can a particle be in a super position of being a hydrogen ion (a proton) and a plutonium atom?

Energy is at the heart of the quantum measurement problem: In order to measure a quantum particle some kind of interaction with the particle must be there. This interaction changes (or not) the state of the particle. It is a bit like this: Suppose I am sitting in my home country and I have to measure the length of some grassfield in Germany or Belgium but I can only use atom bombs for that. No matter how smart I craft my grass length measuring device, the giant explosions from the atom bomb will bring a great uncertainty in the outcome of the measurements… Here is the video:

The cat is also an observer…

Ok, now for the lesser known but rather interesting envelope exchange problem. In a nutshell it goes as next:

You can choose one of two closed invelopes and they contain money. The only thing you are told is that the amount in one of the envelopes is double that of the other envelope.

Now you play the game and you choose one of the envelopes, let’s say it contains 100€. You are asked by the quiz master if you want to keep those 100€ or that you want to change your choice and go for the other envelope.

You think about that for a few seconds and you figure out: If this envelope has 100€ and given the rules of the game, the other envelope contains 50€ or 200€ with equal probability of 50%. Suppose I want to swap to the other envelope, what is my expectation for the amount of money? That is simple, both 50€ and 200€ have 50% probability so the expectation of swapping becomes 0.5*50 + 0.5*200 = 125€. Therefore it makes sense to swap and choose the other envelope.

But hey, whatever envelope you choose at first and you find X money in it, isn’t it weird to swap that always? If you would have chosen the other envelope you would also swap…

This envelope swap problem or paradox has a relative simple solution: You assume equal 50% probabilities for having double or half the amount of money you found in the first envelope. But in that case the whole thing crashes because you are now calculating with three outcomes: the 100€ from the first envelope and two other amounts 50 and 200 Euro while there are only two enveloples. It is unwise to calculate the expectation values because the 50€ and 200€ exclude each other: if the outcome 50€ is observed all of the time the 200€ was non existant. And as such the expectation value makes no sense for an individual experiment.

Ok, let me end this post with a standard wiki around the two envelope thing: Two enveloples problem. https://en.wikipedia.org/wiki/Two_envelopes_problem

End of this post.

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.

Part 20: On the structure of non-invertible 4D complex numbers.

In general it is rather hard to find non-invertible 4D complex numbers because the determinant is non-negative everywhere. Just try it yourself, write down just one 4D complex number that is non-zero and not invertible.

That is not an easy task, after some time you will find some but do you have all?

But if you understand the concept of the eigenvalues that every 4D complex number Z has, it is easy to understand that if a 4D complex number is non-invertible at least one of the eigenvalues must be zero.

In previous posts we already unearthed the four eigenvalue functions that return the four eigenvalues each 4D complex number has.

In this post we will try to find where these eigenvalue functions are zero.

Since eigenvalue functions come in pairs whenever possible in the case of 4D complex numbers we only have two pairs of those eigenvalue functions.

Let’s stop the talking and just post the twelve pictures that make up part number 20 into the basics of the four dimensional complex numbers.

 

 

 

 

 

 

 

 

 

 

 

 

Ok, that was it for part 20 in the series that covers the basics of 4D complex numbers.

See you around my dear reader.

A teaser question: Can you prove this inequality?

Recently I am working on part 20 to the basics of the 4D complex numbers. Ok ok if you need 20 parts to explain ‘the basics’ how basic is it you can ask yourself.

You can argue long and short on this: are fresh Cauchy integral formula’s really ‘basic stuff’? I don’t know how a democratic vote among professional math professors would fall down.

Anyway, an important property of the determinant of 4D complex numbers is the fact that the determinant is always non-negavite. At least it is zero and at those points in space we have found a non-invertible number.

In part 20 on the basics to 4D complex numbers we will look when the eigenvalues of 4D complex numbers vanish; at those points the stuff is non-invertible & that is what we will be hunting on part 20.

In the next picture you see a difficult to understand inequality & the teaser question is:
Can you prove this inequality via math methods that do not use 4D complex number theory at all?

If so, you should definitely pop up a second pint of perfect beer on a late Friday evening.

Ok, that was it. Till updates in part 20 where we try to find all non-invertible 4D complex numbers in a not too difficult way.

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.