Why Complex Numbers as 2X2 Matrices???

I think this video speaks for itself.

[Kydo]

This is tremendous, thank you 🙏 I think with the right compiler your implementation would compute at the same speed as the algebraic implementation. Kudos and thank you for demystifying imaginary numbers for me!

[Quroxify]

Good idea. The algebraic representation doesn't show as much. I like the way you use the word incomensurate. Great analogy with mass and charge.
Interesting nuance about and, vs or. Do those logic operators draw a line of separation in logic.

[assemblyofsilence]

Just off the top of my head without thinking about it: would it be reasonable to say that the Real component might be correlated with DC (scalar) and the Imaginary with AC (polar)??

[Phaze-rbmp]

Excellent work! I'm having some interesting thoughts. I'm thinking of x^2 as x duplicating itself (creation in space). thinking of square root being a transformation operator to counterspace. Its a split. thats why complex numbers are in pairs. Maybe its just a real number/value/thing divided in 2 parts. 1 becomes two or two becomes 1. That is why its imaginary, because separation (into polarity) is an illusion, not ultimately real. lol The creation process going from unity to polarity and back. multiplying those two parts (polarity) i x -i = 1 (wholeness/unity). male + female = a real creation. Or come to think of it, since everything is attracted back to the wholeness/singularity of counterspace, perhaps those imaginary numbers represent wholeness....as you said, not positive, negative or zero....but all of them (superposition).

this makes me think of superposition in the quantum. then isnt the quantum just transforming things from counterspace (imaginary) to space (real). I think going thru the singularity you just fractally arrive at the same thing. Another iteration of the fractal sphere seemingly packed into a point (seed). But entering counterspace would just be the new space and this mirrored inverse reality becomes counterspace? Not sure that all of my associations are valid, but I feel some makes sense.

as genius Chris Langan says that 4D metatime (whole timelines are created there) is orthogonal to our reality/existence. Just makes me think of complex numbers again. Using 90 degree rotations to map that larger set of data, using some kind of complex number transformation into our reality. Havent had time to read his papers, but i believe he mentioned stuff about transformations and matrices and eigenvalues in regards to this mapping.

God dang this stuff is so exciting and yet frustratingly chasing the carrot with no real tangible change no matter how much insight lol. Im gonna be pissed if all of us hard working smart people cant make a major breakthru in the next few decades. :)

[paulstclairterry]

I disagree with most of the comments below. This is an extremely confused and repetitive description which completely misses the mathematical reasoning behind the matrix representation of complex numbers. The representation of the complex numbers as a two-dimensional real vector space reflects an isomorphism of R^2 to C over the field of scalars R. C is also a field and the axioms of a vector space do not support vector multiplication. An exact representation of C as a field is given by presenting a+ bi as the 2x2 matrix R1C1 a, R1C2 –b, R2C1 b, R2C2 a. These matrices, with the exception of the additive identity, (the zero matrix), are invertible with determinant equal to the square of the modulus. Also, matrix multiplication with this form of matrix is (unusually) commutative. Thus, we can look at the complex numbers as vectors (a, b) with basis {1, i} and more comprehensively as linear transformations, specifically rotations with scaling along the Re and Im axes. Since these matrices also form a vector space, we can express all complex numbers as a linear combination of the basis vectors the identity matrix and the rotation anticlockwise through π/2.

[zdenekbreza]

Yes, efficiency vs. complexity (with simple laws behind it– simplex complexity).

Commentary on the end of the video and on various statements on the effectiveness of these calculations. Efficiency does not mean understanding.

Nature works with complexity, Euro-American culture with efficiency (not people in general, although now this applies to almost everyone, because the Euro-American view has been enforced more or "less" voluntarily across the planet).
Complexity - stability, resilience, more elements are taken into account, cooperation prevails over competition, slower, ...
Efficiency - one thing is chosen and the rest go aside - everything else must die because of it, everything else is intentionally "overlooked". It is unstable, very fragile, but also very powerful, fast - growth and Fall, ... palm oil "fields", corn fields, only one theory possible, only one religion possible, only one race possible, only one opinion possible, I wanted to write one gender, but here the pendulum swung to the other side in last years-I am from Europe and this issue is perceived a little differently here, so if it is too politically incorrect statement, I will delete it, because politics does not belong here anyway – although...

And this way of thinking effectively is, of course, reflected in everything. Obviously, even in reflecting on what equations tell us at all, and not only that, but how we create equations, it reflects on how we think about the world, what we come up with in theory (a change in thinking will change theories and change some "natural laws ", at least what we think are laws), what we are creating for the technologies that are supposed to save us (that's ridiculous!! we absolutely omit people from those equations about salvation through technology - who make them, who use them !!The camera surveillance and the submachine gun won't hurt you, the man/woman/it who is sitting, standing behind it, he/she/it could. How is it supposed to change when everyone is still concentrating on the outside and not looking at what they are like?)

Note on complex numbers:
In a sense, the real component of a complex number could correspond to a transverse wave (because it is either positive or negative) and an imaginary component to a longitudinal (standing?) wave, because it has both components (positive and negative) at the same time, right?

[rogerconnolly]

If complex numbers are about rotation. Is there any priority over the direction of rotation, ie clockwise or counterclockwise. I ask this as multiplication of matrices seems to relate to clockwise yet this “i” matrix of yours seems to suggest that it is about counterclockwise. I am not sure what I am looking for here. Maybe the answer is that nature functions or expands counterclockwise.

[TheWestman]

Image on left side, what's this? How you get it?

[mes]

Epic work Lori! Your videos inspired me to make my own video relating the complex plane to rotation matrices. Check it out on my channel if you want (it's a bit long since it's a derivation video haha). :)