For me it was a strange experience to construct a square matrix and if you take the determinant of the thing the lower dimensional volume of some other thing comes out. In this post I will calculate the length of a 3D vector using the determinant of a 3×3 matrix. Now why was this a strange experience? Well from day one in the lectures on linear algebra you are thought that the determinant of say a 3×3 matrix always returns the 3D volume of the parallelepiped that is spanned by the three columns of that matrix. But it is all relative easy to understand: Suppose I have some vector A in three dimensional space. I put this vector in the first column of a 3×3 matrix. After I add two more columns that are both perpendicular to each other and to A. After that I normalize the two added columns to one. And if I now take the determinant you get the length of the first column. That calculation is actually an example below in the pictures.
Well you can argue that this is a horrible complicated way to calculate the length of a vector but it works with all parallelepiped in all dimensions. You can always make a non-square matrix, say an nxd matrix with n rows and d columns square. Such a nxd matrix can always be viewed as some parallelepiped if it doesn’t have too many columns. So d must never exceed n because that is not a parallelepiped thing.
Orthogonal matrices. In linear algebra the name orthogonal matrix is a bit strange. It is more then the columns being orthogonal to each other; the columns must also be normalized. Ok ok there are reasons that inside linear algebra it is named orthogonal because if you name it an orthonormal matrix it now more clear that norms must be one but then it is more vague that the columns are perpendicular. So an orthogonalnormalized matrix would be a better name but in itself that is a very strange word and people might thing weird things about math people. Anyway loose from the historical development of linear algebra, I would like to introduce the concept of perpendicular matrices where the columns are not normalized but perpendicular to each other. In that post we will always have some non-square matrix A and we add perpendicular columns until we have a square matrix.
Another thing I would like to remark is that I always prefer to give some good examples and try not to be too technical. So I give a detailed example of a five dimensional vector and how to make a 5×5 matrix from that who’s determinant is the length of our starting 5D vector. I hope that is much more readable compared to some highly technical writing that is hard to read in the first place and the key idea’s are hard to find because it is all so hardcore.
This small series of posts on the Pythagoras stuff was motivated by a pdf from Charles Frohman, you can find downloads in previous posts, and he proves what he names the ‘full’ theorem of Pythagoras via calculating the determinant of A^tA (here A^t represents the transpose of A) in terms of a bunch of minors of A. A disadvantage of my method is that it is not that transparant as why we end up with that bunch of minors of A. On the other hand the adding of perpendicular columns is just so cute from the mathematical point of view that it is good to compose this post about it.
The post is eight pictures long so after a few days of thinking you can start to understand why this expansion of columns is say ‘mathematical beautiful’ where of course I will not define what ‘math beauty’ is because beauty is not a mathmatical thing. Here we go:
Inside linear algebra you could also name this the theorem of the marching minors. But hey it is time to split and see you in the next post.
A few posts back I showed you that pdf written by Charles Frohman where he shows a bit of the diverse variants of the more general theorem of Pythagoras there is. At school you mostly learn only about the theorem for a triangle or a line segment and it never goes to anything else. But there is so much more, in the second half of this post I show you three vectors in 4D space that span a parallelepiped that is three dimensional. From the volume of such a thing you can also craft some form of Pythagorean theorem; that paralellepiped can be projected in four different ways and the squares of the four volumes you get equals the square of the original parallelepiped. I would like to remark I hate that word ‘paralellepiped’, if you like me often work without any spell correction this is always a horrible word…;)
Now my son came just walking by, he reads the title of my post and he remarks: It sounds so negative or sarcastic this ‘full theorem’. And no no no I absolutely do not mean this in any form of negative way. On the contrary I reconmend you should at least download Charles his pdf because after all, it was better compared to what I wrote on the Pythagoras subject about 10 years ago.
But back to this post: What Charles names the full theorem of Pythagoras is likely that difficult looking matrix expression and from the outside it looks like you are in a complicated space if you want to prove that thing. The key observation is that all those minor matrices are actually projections of that n x k matrix you started with. So that is what the first part of this post is about.
The second part is about a weird thing I have heard more than once during my long lost student years: Outside the outer product of two vectors we have nothing in say 4D space that gives a vector perpendicular to what you started with. I always assumed they were joking because it is rather logical that if you want a unique one-dimensional normal vector in say 4D space, you must have a 3D dimensional thing to start with. That is what we will do in the second part: Given a triple of vectors (ABC) in four dimensional space, we construct a normal vector M to it. With this normal vector, after normalization of course, it gives now a handy way to calculate the volume of any of those paralellepiped things that hang out there.
Ok, lets go: Six pictures long but easily readable I hope.
All that is left is trying to find back that link to Charles his pdf.
Last week I started thinking a bit about that second example from the pdf of Charles Frohman; the part where he projected a parallelogram on the three coordinate planes. And he gave a short calculation or proof that the sum of squares of the three projected areas is the square of the area of the original object.
In my own pdf I did a similar calculation in three dimensional space but that was with a pyramid or a simplex if you want. You can view that as three projections too although at the time I just calculated the areas and direved that 3D version of Pythagoras.
Within the hour I had a proof that was so amazingly simple that at first I laid it away to wait for another day or for the box for old paper to be recycled. But later I realized you can do this simple proof in all dimensions so although utterly simple it has absolutely some value. The biggest disadvantage of proving more general versions of the theorem of Pythagoras and say use things like a simplex is that it soon becomes rather technical. And that makes it hard to read, those math formula’s become long can complex and it becomes harder to write it out in a transparant manner. After all you need the technicalities of your math object (say a simplex or a parallelogram) in order to show something is true for that mathematical object or shape.
The very simple proof just skips that all: It works for all shapes as long as they are flat. So it does not matter if in three dimensional real space you do these projections for a triangle, a square, a circle, a circle with an elleptical hole in it and so on and so on. So to focus the mind you can think of 3D space with some plane in it and on that plane is some kind of shape with a finite two dimensional area. If you project that on the three coordinate planes, that is the xy-plane, the yz and xz-plane, it has that Pythagoras kind of relation between the four areas.
I only wrote down the 3D version but you can do this in all dimensions. The only thing you must take in account is that you make your projections along just one coordinate axis. So in the seven dimensional real space you will have 7 of these projections that each are 6 dimensional…
This post is four pictures long, I did not include a picture explaining what those angles alpha and theta are inside one rectangular triangle. Shame on me for being lazy. Have fun reading it.
So all in all we can conclude the next: You can have any shape with a finite area and as long as it is flat it fits in a plane. And if that plane gets projeted on the three coordinate planes, the projected shapes will always obey the three dimensional theorem of Pythagoras.
Ok, thanks for your attention and although this inner product kind of proof is utterly simple, it still has some cute value to it.
A few months back I found a very good text on more general versions of the good old Pythagorean theorem. Since in the beginning of this text the author Charles Frohman did the same easy to understand calculations as I did a long time ago I more or less trust the entire document. But I did not check the end with those exterior calculations, I don’t know why but I dislike stuff like the wedge product. The second pdf is from myself, likely I wrote it in 2012 because a proof of a more general version of the theorem of Pythagoras was the first math text I wrote again after many years. After that at the end of 2012 I began my investigations into the three dimensional complex numbers again and as such this website was needed in 2015.
Anyway I selected 3 details from these two pdf’s that I consider beautiful math ideas where of course I skip a definition of what ‘beautiful’ is. After all the property ‘mathematically beautiful’ is not a mathematical object but more a feeling in your brain.
Let me start with four pictures where I look into those 3 selected details, after that I will hang the two pdf texts into this post.
The first pdf is from Charles Frohman. May be you must download it first before you can read it, I should gain more experience with this because the pdf format is such a hyper modern development…;) The first text is from 2010:
(Later I saw there were some old notes at the end of my old pdf, you can neglect that, it has nothing to do with the Pythagoras stuff.)
There is little use in comparing these texts, I only wanted to make a proof that uses natural induction so I could prove the theorem in all dimensions given the fact we have a proof (many proofs infact) for the theorem of Pythagoras with a rectangular triangle. Charles his text is more broader and the main piece is the proof for that determinant version of the theorem of Pythagoras.
At last a remark about the second detail of mathematical beauty: Charles gave the example of a parallellogram where the square of the area equals the sum of squares of the three projections on the three coordinate planes. I think you can take any shape, a square of a circle it does not matter. It only matters it is a flat thing in 3D space. After I found that within the hour I had a proof for the general setting of this problem in higher dimensional real space, may be this is for some future post.
For the time being let us split and go our own ways & thanks for the attention of reading this.