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.

[marcushendriksen]

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.

[josephkalman]

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

[quentintaylor]

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

[ramlarizvi]

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.

[jacobhawthorne]

The single most underrated math channel ever.

[scottgoodson]

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.

[mattyedlin]

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!

[DatBoi_TheGudBIAS]

this will be useful to solve imaginary systems in my calculator dat doesnt support imaginary coeficients

[alexrosellverges]

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

[yurigansmith]

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

[yairraz]

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...

[sdmartens]

the quality of your videos is breathtaking

[persanazh]

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

[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?)

[avtaras]

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

[dr.rahulgupta]

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

[paulisaac]

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

[pianavela]

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

[gabrielkrivian]

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

[duartesilva]

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