The strange utility of this mutant matrix. 

Well, you could consider the split-complex numbers which have 1^2 = 1 and j^2 = 1 as basis elements. Removing those we see that the remaining elements do produce an even split (the same is true of split quaternions, etc.). Of course if we had started with 1^2 = 1 and i^2 = -1 then we would get the normal complex/quaternions/octonions.

Yes, it is. You have for any couple of coefficient the sum and the differencein the matrix, and so you can easily solve and prove uniqueness

Yes. The Octonians are obviously 8-dimensional over |R, hence their name. For the number-vector matrix representation, if you count the "number entries" in those whacky matrices, there are the 2 regular numbers on the main diagonal, plus one 3-dimensional vector each on the two off-diagonal entries, which gives you the remaining 3+3 = 6 dimensions for the total of dim = 2+6 = 8. So the dimension is already correct, and it's pretty easy to pick numbers (1s and 0s should suffice) using that map which will give you 8 linearly independent vector-matrices. So the dimensionality works out which proves the bijection. More concretely, ignoring the matrix shape and just using standard linear algebra for the 8 coefficients, it will even be easy enough to write up the inverse map. On terminology, I guess the term "representation" ought to always imply an isomorphism (my memory may be a bit rusty here), and an isomorphism is certainly always bijective (that's part of its definition).

@@franzlyonheart4362 Thank you for this clarification, and also a line from a user above, about it having sum and difference incorporated, so it's possible to extract coefficient for e's.

If i go off other comments - sedenions have 16 degrees of freedom, while the 2x2 matrix would have only 12 diffirent degrees of freedom.

No, this is the structure of split-octonions, otherwise it would be just normal octonians

This is "split-octions." If they have the same orientation, then it would be regular octonians.

@@assassin01620 in regular octonions, all the e's square to -1. The change is some now square to 1 instead - I thought that was the only change... I'm sure the diagram should change

oops meant to say "not sure" the diagram should change

Actually, we are both wrong - I just looked it up. That is the correct diagram for regular octonions - not sure why it has that wart. So, the diagram doesn't change for split octonions, just what squares to -1.

I don't really think this definition is "natural" in any sense, it's not like we use these matrix-vectors elsewhere. And cross product only existing in 3/7 dimensions means it's not very generalizable (though the 7 dimensional one is roughly giving you the octonions already). I imagine Zorn was just messing around with how to add non-associative operations on matrices and found this.

These octonians and split version are interesting to me. I was doing all that group and ring theory stuff, and that Lie algebra stuff, myself earlier in my life. I work in a totally different field now, but it's good fun to watch such quirky new things like "Octonians" on RU-vid. I hadn't even come across those back in my mathematician former life (everyone learns about quaternions in the first semester, but not everyone hears the stuff beyond). Having said this, I totally forgot how the cross product works. Haha, perhaps it's too simple!

The definition of matrix multiplication using cross product inside entries like that has a nice property: if you replace the R^3 vector with an R (scalar), you get normal matrix multiplication because cross-product on scalars is always 0, so they vanish from the sum. (I deduce that the cross-product of scalars is 0 from something mentioned in an earlier video: 0, 1, 3, and 7 dimensional cross-products are the same as multiplication of the reals, complex, quaternions, and octonions respectively, if you *only* use the imaginary parts and keep setting the real parts to 0. Specifically, scalar cross-product is related to the multiplication of imaginary parts of complex numbers, which always results in a 0 imaginary part.)

one way is to say R -> Q -> Z -> N, but in algebraic geometry some would consider the tropical numbers as the 'next step' after C and R

I imagine by first writing split-octonion multiplication in terms of dot products and cross products (you can do the same thing for quaternion multiplication) and then noticing that the resulting formula can be written in terms of a "matrix" product if you squint enough.

Mathemagicians run the world.

We pure mathematicians wear that WTF badge proudly! It's what we're all about.

@@jimdotz They laugh at us now, they always do... but we get the last laugh because we know about limits.

It's not a group, because it's not associative.

No, since it contains elements without inverses. Indeed since matrix multiplication is associative we know that no matrix representation can exist.

No … I presume you are attempting to answer his "homework question" why no (isomorphic) Matrix representation exists. The reason is the fact that matrix multiplication is always associative, but Octonian multiplication is non-associative.