Complex numbers as matrices | Representation theory episode 1

  Рет қаралды 35,915

All Angles

All Angles

Күн бұрын

Пікірлер: 64
@pedroth3
@pedroth3 3 ай бұрын
I have always known that complex numbers could be represented as matrices, but never stopped to think about their application to matrix theory, Perfect video
@zyklos229
@zyklos229 2 ай бұрын
now try the same with quaternions :P
@johnnavarra9037
@johnnavarra9037 Ай бұрын
3T4A4s
@jarahfluxman20
@jarahfluxman20 3 ай бұрын
The reason complex conjugation corresponds to transposition in the matrix rep is because the inner product :=z* w on the complex plane is mapped to the standard dot product on R^2, and complex conjugation on C and transposition on R^2 both represent the hermitian adjoint.
@AllAnglesMath
@AllAnglesMath 3 ай бұрын
That's such a cool connection! I have been thinking about making a video about adjoints for a while, but I haven't gotten around to it yet.
@superuser8636
@superuser8636 2 ай бұрын
Shout out to all my hermitians
@dan0_0nad76
@dan0_0nad76 3 ай бұрын
Bro i swear i love you
@dan0_0nad76
@dan0_0nad76 3 ай бұрын
Just failed an exam and got so frustrated that i almost felt like i should not be doing math ever again. Your intuitive explanations really keeps me motivated
@AllAnglesMath
@AllAnglesMath 3 ай бұрын
@@dan0_0nad76 Sorry to hear that you had a bad time with an exam. It happens to the best of us. I hope you'll be able to keep doing what you love.
@johnstuder847
@johnstuder847 2 ай бұрын
Wow - thank you! So glad there are people like you that understand this stuff well enough to combine seemingly disparate concepts and provide a high level summary without the details and rigor that obscure many of the important concepts. Well done! I kept hoping you would work my favorite subject in there - how the Fourier transform, SVD, (singular value decomposition), tensors, and Roots of Unity are related. Keep up the great work! Love your channel.
@BlueDog15391
@BlueDog15391 2 ай бұрын
Sounds cool, where would you recommend reading about the connection between Fourier transform and SVD for those who are impatient?
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
I can see the connection between Fourier and roots of unity, but where do tensors and the SVD come in? If you have a good reference where I can learn more, I will definitely consider making a video about that. Right now, you might be interested in a video we have on Patreon that looks at Fourier analysis as the study of the characters of a circle.
@maxwellmogadam399
@maxwellmogadam399 3 ай бұрын
this is my personal favorite subject, a fun exercise is to think about complex valued matrices, for example try looking at a pauli matrices as a set of nested matrices
@peskarr
@peskarr 3 ай бұрын
heavily underrated content, instant subscription
@TurboLoveTrain
@TurboLoveTrain 2 ай бұрын
This should be taught day one of any linear equations curriculum.
@mrervinnemeth
@mrervinnemeth 3 ай бұрын
Fun fact: i is an anti-symmetric matrix here. You can decompose every square matrix to the sum of a symmetric and an anti-symmetric component every time: ½(A + Aᵀ) is the symmetric and ½(A - Aᵀ) is the anti-symmetric component, where Aᵀ is the transposed matrix. This doesn't add up entirely because a symmetric matrix is not necessarily a multiple of the identity matrix, but it still looks like decomposing a complex number to a real and an imaginary component. Stupid me, I had to correct the formulas.
@mariotabali2603
@mariotabali2603 2 ай бұрын
This is rather good so far
@rylieweaver1516
@rylieweaver1516 2 ай бұрын
All angles never disappoints 🙌🏻
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
Thanks for having so much faith in our channel.
@SkielCast
@SkielCast 2 ай бұрын
This series looks really promising! It's great! I would suggest to speed up a little bit the narrative, I had to listen it at 1.5 to keep me engaged
@eqwerewrqwerqre
@eqwerewrqwerqre 3 ай бұрын
Wow, this is amazing. I've got go watch all your videos real quick.
@AllAnglesMath
@AllAnglesMath 3 ай бұрын
Feel free to ask questions in the comments if needed.
@apuji7555
@apuji7555 3 ай бұрын
This is super interesting!
@sleepysnekk
@sleepysnekk 3 ай бұрын
honey wake up new allangles series 🗣️🗣️🔥🔥
@MathsSciencePhilosophy
@MathsSciencePhilosophy 3 ай бұрын
Nice idea to write i as a matrix ⭐❣️
@FaTeAnimationVault
@FaTeAnimationVault 2 ай бұрын
In computational physics this is a very, very useful tool.
@finite1731
@finite1731 3 ай бұрын
This is bascially the reason why z* doesnt have a derivative since the jacobean of the transformation is a reflection and not a rotation and scalling (the only transformations possible when multiplying by a complex number) which is effectively the reasoning behind the Cauchy-Riemann equations, it took me ages to understand the link between the linear transformation due to complex multiplication and the restricted set of matricies which describe complex multiplication which the jacobean must be in, since it too must reprosent complex multiplication thus must be of this form thus the CR equ.'s. (whenever I found the jacobean, I assumed that that must mean a derivavtive must exist, but no)
@finite1731
@finite1731 3 ай бұрын
You video made this idea very clear, well done
@mrervinnemeth
@mrervinnemeth 3 ай бұрын
Geometric algebra gives you a hint how to do this properly. Let's go to G(2,0), the two-dimensional space. Your i matrix is simply a transformation, and it's a bivector in GA, and a pseudoscalar in this space. We have the x axis and the y axis, but y can be written simply as y = i x. From now on we can describe every point on the plane with a x + b i x. This gives an algebra entirely isomorphic to the complex numbers. Or we can choose G(0,1), which is much simpler, and still isomorphic to complex numbers.
@AllAnglesMath
@AllAnglesMath 3 ай бұрын
As it happens, I'm preparing the series on GA right now. And you've hit the nail on the head. We will publish 2 videos about this later. Also, G(0,1) is trivially isomorphic to C, but I prefer the way they emerge from G(2,0) because it's much more surprising. You define 2 elements that square to 1, and magically you get a new element that squares to -1. Math is full of such surprises.
@mrervinnemeth
@mrervinnemeth 3 ай бұрын
@@AllAnglesMath Sorry for the spoiler :)
@DandapaniTripaathi
@DandapaniTripaathi 13 сағат бұрын
Just helped me in my group theory course, thank you
@mjkhoi6961
@mjkhoi6961 2 ай бұрын
I always wondered why the "i" matrix had the negative sign in the top left instead of the bottom right, this explains it perfectly
@AlanCanon2222
@AlanCanon2222 2 ай бұрын
I learned about complex numbers while programming a Mandelbrot set generator in Pascal on my parents' PC, based on an algorithm published in Scientific American. I remember getting pretty deep into complex numbers, and proving what I later knew to be famous basic theorems. Namely, if an equation is true among complex numbers, then it is also true among their complex conjugates; and that if you multiply two complex numbers represented as vectors, the rotation of the product is the sum of the rotations of the factors, and the length of the product is the product of the lengths. Of course now I know my way around the complex plane in terms of r* e ^ (i * theta), which makes these facts self evident, but I remember proving them just using algebra.
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
That's pretty cool! When you proved that second theorem, especially the part where the angle of the product is the sum of the angles, did you realize that you had discovered an exponential relation?
@OmarAhmed-ic4fw
@OmarAhmed-ic4fw 2 ай бұрын
What an Amazing video I hoped not to end!!
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
Thank you! Sorry that it had to end 🤷
@OmarAhmed-ic4fw
@OmarAhmed-ic4fw 2 ай бұрын
@@AllAnglesMath It would be great to see another video in this series soon!
@markhamann8030
@markhamann8030 2 ай бұрын
This is great. I've been trying to understand capacitance and inductance without resorting to to the square root of -1, which makes the math work but is far from intuitive. This is taking me into a good direction.
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
You can often replace complex numbers by something else (such as matrices or 2D vectors). The math still works, but it's often more verbose. You no longer package the 2 parts into a single number, which leads to twice as many equations. By packaging those back into vectors/matrices, you may be able to recover the original elegance.
@markhamann8030
@markhamann8030 2 ай бұрын
@@AllAnglesMath Yes. The thing I like about representing impedance as [[R, Z][-Z R] is that it's 2 vectors: the top one being the impedance, and the bottom one being a vector with the same magnitude but rotated exactly 90 deg counter-clockwise. So you do V*[[cos t, sin t] [-sin t, cos t]] = [[R, Z][-Z, R] * I*[[cos t, sin t] [-sin t, cos t]] . This shows exactly how the real and reactive elements turn real and 90 deg out of phase currents into real and out of phase voltages while reactive elements turn real and out of phase currents into out of phase and real voltages. It's all quite intuitive when you do it this way. It doesn't depend on an understanding that multiplying by i means rotation.
@nishchayy
@nishchayy 2 ай бұрын
Great work great video. Thank you
@MaxPicAxe
@MaxPicAxe 3 ай бұрын
Wow that's awesome
@brofessorsbooks3352
@brofessorsbooks3352 2 ай бұрын
Amazing keep it up!!
@manfredbogner9799
@manfredbogner9799 2 ай бұрын
Sehr gut
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
Vielen Dank!
@friedrichfreigeist3292
@friedrichfreigeist3292 2 ай бұрын
Please note that i, sin and cos are not variables. This is why one should not write them cursive. This topic is so based. I had a lot of representation matrices stuff in a group theory course.
@blue5659
@blue5659 2 ай бұрын
Is it possible to represent signed numbers in terms of just unsigned numbers in vector form?
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
I'm not sure. I like this question though, it makes me think.
@woomygfx
@woomygfx Ай бұрын
these matrix representations reminds me a lot about functors, could there maybe be a correlation or could be that I'm just confusing things...?
@AllAnglesMath
@AllAnglesMath Ай бұрын
Every matrix representation is a homomorphism; and those in turn are studied as the "arrows" in category theory. Functors are an arrow between such arrows. The condition for functors is indeed very similar to the conditions you see for many other kinds of arrows such as isomorphisms, linear transformations, or diffeomorphisms.
@vishnuvudhisen1874
@vishnuvudhisen1874 2 ай бұрын
Eigen value =i?
@peterfireflylund
@peterfireflylund 2 ай бұрын
What is “number samba”?
@AllAnglesMath
@AllAnglesMath 2 ай бұрын
That's just the phrase I use for the product between two matrices, or between a matrix and a vector. I adopted this phrase half jokingly in the series on linear algebra.
@peterfireflylund
@peterfireflylund 2 ай бұрын
@@AllAnglesMath oh. I hoped I heard misheard something less like a secret club handshake/wink.
@05degrees
@05degrees 3 ай бұрын
Ah, among other stuff this introduction reminds me how I derived quaternion matrices by myself, not without errors at first.
@AllAnglesMath
@AllAnglesMath 3 ай бұрын
Nothing ever happens without errors at first 😉
@TurboLoveTrain
@TurboLoveTrain 2 ай бұрын
The little hole you punched in the middle is why phi doesn't start at zero.
@anamariatiradogonzalez
@anamariatiradogonzalez 2 ай бұрын
Que ne lo paguen que estiy jasta el cñ moño de abajo😊
@hyperduality2838
@hyperduality2838 2 ай бұрын
Real is dual to imaginary -- complex numbers are dual. "Always two there are" -- Yoda.
@omipial2084
@omipial2084 22 күн бұрын
1:11-;6:46-6:56;8:10-8:22;9:03-9:15;
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