Complex Numbers as Matrices

In this video, we'll learn how to view a complex number as a 2x2 matrix with a special form. We'll also see that there is a matrix version for the number 1 and a matrix representation for the imaginary unit, i. Furthermore, the matrix representation for i has the defining feature of the imaginary unit in that it squares to -1. We'll also explore other features of complex numbers that tie into matrix operations, such as the determinant and transpose. Euler's formula even has a sensible interpretation in matrix algebra and we'll see an example of a matrix exponential.

[@marcushendriksen8415]

Dude, your channel is pure gold! I can't believe I'm only just finding it now. I haven't checked out your actual page yet, but I hope you're still making videos.

[@jacobhawthorne1997]

The single most underrated math channel ever.

[@Math_oma]

+Brody Haha, I guess that's slightly better than being the most overrated channel ever.

[@josephkalman9468]

very good, no mention of difficult chidhood, social injustice, and/or any lingering frustration

[@ramlarizvi4732]

This is great! I hv struggled a lot to understand why a complex number should be represented as a matrix in this form and in the process I posted this question on forums such as Quora too .... but trust me the answers were either densely technical or were unconvincing for me. Thanx for making it so clear.

[@yurigansmith]

I really love this approach, especially the version of Euler's theorem with complex numbers as 2x2 matrices.

[@yairraz6067]

Man this video finally cleared up so many concepts that were blurred for me . I hate to say this but the treatment of these concepts in many videos is so confusing . Thank you...

[@danielpaz7696]

you have absolutely no idea how much this helps me, thank you very much

[@catherinejeffares3480]

Thank you! So clearly explained.

[@alexrosellverges8345]

Awesome video, it actually helped me a lot, I am so glad i found it, thanks!

[@mattyedlin7292]

this is an excellent video! I am teaching Deep Learning and introducing automatic differentiation using dual numbers. We can map the calculations anbd pedagogy from this video to explaining dual numbers and their geometric interpretation! A million thank yous!

[@Matematica_Aplicada]

Well done! Really nice explanation!

[@pianavela]

Very nice video! Compliments. The connection with Pauli matrices can help understanding better the matter.

[@vinaykumardaivajna2688]

Very well explained....👌👌

[@sdmartens22]

the quality of your videos is breathtaking

[@Math_oma]

+Shannon Martens That could either be breathtakingly good or breathtakingly bad, haha.

[@dr.rahulgupta7573]

Excellent presentation of the topics. Many many thanks. DrRahul Rohtak India

[@MrCigarro50]

Great video....thanks.

[@may8049]

This Guy is a Legend!You deserve 1M subs

[@Logicallymath]

phenomenal stuff

[@gamingwithdoremon6155]

It's really nice. I recently learned abt the matrix represntation of complex numbers and wondered why one would want to use this form. The properties of such a representation are well explained here. However I would love to see a bit more elaboration on motivation and intuition for this depiction of complex numbers.

[@Robocat754]

Hermitian Matrices. This is the reason I'm here. Knowing what the complex number as a matrix form is can help me understand why I need to take the conjugate when transpose it.

[@paulisaac3489]

Bamn, I'm gonna solve some problems! Love it, subbed, thanks!

[@gabrielkrivian8453]

The very first image literally answered half my questions, and it only took 15 seconds.

[@scottgoodson8295]

Could you make a video about dual numbers (which are of the form a+bε where ε^2=0 but ε does not equal 0)? They seem to have similar properties to complex and split-complex numbers, but there isn't very much information about them.

[@Math_oma]

Sounds like my kind of video. I know only a little about them but I do know that they can also be understood by matrices. I'll do some more research and make a video on this (probably this weekend).

[@ingiford175]

I know I am posting to a really old post, but Micheal Penn has 2 videos that touch up on it: The strange cousin of the complex numbers -- the dual numbers. The complex number family.

[@MS-cj8uw]

Love you ...thank you

[@dayeol_han]

Thank you!

[@dimitrisdimitriou6969]

thanks a lot!

[@pavelperina7629]

How to deal with the fact that complex number is sometimes vector and sometimes matrix? I can somehow understand that it is actually ordered pair of number (a,b) and addition is the same as vector addition whereas multiplication by (c,d) is defined as: take (a,b) and scale it by c. Than rotate (a,b) by 90deg and scale it by d. Then add these vectors together. That means complex number behaves more a like transform that can be written as matrix, but out of sudden it's little bit confusing to represent it as both 2x2 and 2x1 matrix. Another confusion is that it's more usual to use vectors for rotation and scaling as c=crossprod(a,b) and not as vector (1,0) maps into z=(a,b).

[@AdoptedPoo]

goated channel

[@67persanazh]

the eigenvalues of the matrix [[a,-b],[b,a]] are a±bi! remarkable

[@austinosogwin4121]

very helpful

[@duartesilva7907]

You should look at Pauli matrices. He has all you showed, minus the one about a + bi and i^2.

[@omargaber3122]

Thank you

[@MagicGonads]

How would you use the complex exp definition to work for exponentiation? For real numbers it's b^n = exp(n*ln(b)), is the same true for complex numbers? Also how would you calculate the ln of a complex number? I'm working on a complex class in python (reinventing the wheel) but I'm stuck in a rut trying to figure out how to code in an exponential function that allows you to raise a complex number to a complex power (I'll accept the first root of any power, even if there's infinitely many)

[@Math_oma]

+Magic Gonads We can extend the Taylor series normally used for real numbers to matrices: exp(X)=1+X+X^2/2!+X^3+3!+... But now you interpret "1" as the identity matrix and X is a matrix. Some of the properties of exponentiation are going to fail to hold because matrix multiplication does not commute in general. I'd look into the matrix logarithm - a lot has been written on it. Calculating logarithms for complex numbers can be naively thought of as real numbers, by asking "the exponential of what number gives me i", for example. One possible answer is pi/2*i. However, there are some subtleties which are the topic of complex analysis in terms of making this stuff work.

[@MagicGonads]

Mathoma Ok thanks, at the moment I'm not too fussed about what the real answer is as long as the function is accurate enough and continuous, particularly for log spirals and mandelbrot-like sets. I remember using the (can't remember the name of it) theorem that said to raise a complex number to an integer power (that it wasn't perfectly defined for non-integers) the equation was r^n * (cos(nt)+isin(nt)) for complex numbers in polar form. But clearly, that is for raising complex to real but I want to try complex to complex, because I'm sure it will lead to beautiful images.

[@ZAUpdates]

I think you made a boo-boo at 14:13 . Since Θ is the scalar part in Θi, shouldn't the derivative of e^(Θi) = Θ * e^(Θi) ?

[@Math_oma]

+OZW1N9 But notice there I was taking the derivative with respect to Θ.

[@ZAUpdates]

Ah, I see. Clearly I'm too used to putting my scalar on the left of the variable. Thanks for the great vids! I've literally spent my entire Sunday learning from your videos :)

[@petermahoney4518]

in demonstrating that ac-bd+(ad+bc)i has the matrix form (3:25), where has i gone?

[@billygraham5589]

I have the same question, did you ever find out?

[@ApplepieFTW]

I still don't get why you can convert the two. Isn't it true that in that matrix a and c (so a,b) record the location of i-hat and b and D (so -b, a) record the location of j-hat? So how come you should be allowed to talk about a complex number, just a point in the plane, as a linear transformation/matrix? I don't understand. (also, why can you "discard" the factorised i? Because it is the vertical part? Still, how is a number suddenly a linear transformation?)

[@MrPetoria33]

ApplepieFTW The complex numbers are what’s called a field. It can be shown that all fields are vector spaces over themselves (all vector spaces are defined over a field). Therefore, the complex numbers are also a vector space. Any complex number is also map from the vector space of complex numbers into itself (a complex number times another complex number is still a complex number; reals are a subspace). To be a linear transformation, this map must satisfy the superposition principle. Do complex numbers satisfy this requirement? Yes, z(a*z1 + b*z2) = a*z(z1) + b*z(z2) where z, z1, z2, a, b are all complex numbers. Therefore, complex numbers are linear transformations.

[@quentintaylor1016]

why is a+bi represented as a matrix but c+di is represented as a vector , shouldnt they both be represented as a matrix?

[@avtaras]

Very detailed explanations. Bye bye 3Blue1Brown, hello Mathoma :)

[@pavelperina7629]

I would not go that far. 3B1B has nice visualizations, that are sometimes handy and sometimes it's good to have different views on the same topic. Sometimes it would be better to have paper (html page, PDF) that is commented in video and that is free to download. Printed version has advantage of writing down notes and possible cross references. Now I'm trying to understand quaternions and it's horrible to track down what is well explained in which video and getting greater picture. Sometimes some video starts making perfect sense later. And it's horrible to realize how little I know about complex numbers.

[@lorostotos5647]

both are great

[@soupman55]

at 10:19 should write -(b)(-b)

[@maxwellsequation4887]

Woah [insert mind blown meme here]

[@LoveREvolution1111]

a × d + bi

[@LoveREvolution1111]

i = c3