How (and why) to raise e to the power of a matrix | DE6

General exponentials, love, Schrödinger, and more.
An equally valuable form of support is to simply share some of the videos.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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The Romeo-Juliet example is based on this essay by Steven Strogatz:

The book shown at the start is Vladimir Arnold's (excellent) textbook on ordinary differential equations.

Need a review of ordinary powers of e?

Or of linear algebra?

Timetable
0:00 - Definition
6:40 - Dynamics of love
13:17 - Linear systems
20:03 - General rotations
22:11 - Visualizing with flow

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Code for this video:

These animations made using a custom python library, manim. See the FAQ comments here:

You can find code for specific videos and projects here:

Music by Vincent Rubinetti.

Download the music on Bandcamp:

Stream the music on Spotify:

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Various social media stuffs:

[carl]

Publishing math content on April Fool's Day is a really good way to have your audience evaluate your work critically.

[freddyfozzyfilms]

When we needed him the most, he returned.

[Quotenbrtchen]

Juliet: "It just seems like your feelings aren't real"
Romeo: "I think your doubts are only imaginary"

[maasbekooy]

In the beginning: Damn this one is half an hour.
At the end: it can't be over already. I want more!

[wxcvbndu]

"Fun little exercise" and "Ordinary differential equations textbook" are two groups of words I didn't expect to ever hear in the same sentence again now that I'm done with uni

[xjdfghashzkj]

I've been a big fan of the Calvin and Hobbes comic strips since I was about ten years old. One storyline began with Calvin receiving a mysterious letter in the mail that appeared to be from some kind of secret agent -- the text consisted of individual letters clipped and pasted from magazines, the return address was nothing but a skull and crossbones, the writer advised Calvin that a coded message would soon follow -- all the tropes. The strip ends with Calvin sprinting off, yelling "This is so cool I have to go to the BATHROOM!" I found that response to be both hilarious and relatable. Calvin had no idea what was going to happen next, nor if he would even be able to figure any of it out -- he just knew it was so bizarrely exciting that he had to drop everything he was doing and hope to make sense of it just for the sheer joy of it.

Anyway, that's pretty much exactly how I feel whenever I hop onto youtube and see that 3Blue1Brown has posted a new video.

[kevinlee]

As a physics student, I'm really grateful for being able to know this channel and all your intuitive videos.

[onusbaum]

Please !!! Do the video on the Schrödinger equation: you are the only one out there, who could truely explain it and its beauty 💖💗💖

[aaronbalan]

There is NO person that I know that i have not recommended this beautiful channel to. You are my hero and icon. :)

[joaofrancisco]

I just cannot be grateful enough to live in a time when this kind of content is freely available on the internet. Thank you for all your work, Grant.

[emlun]

I just love this intro, pointing out that the notation is initially nonsense but then receives a meaning we choose to give it. I will always remember how one of my professors introduced distributions:

"Alright, we've hand-waved a definition of the Dirac delta-"function" which is "infinite" in one point and zero everywhere else. That makes no sense, a function can't do that. But here's what we're going to do: we're going to write an integral over this "function", then we're going to _cheat, _ and then we'll end up with something that makes sense in the end. Right now, this integral is nonsense. So let's instead _define_ a meaning for this integral. And that's it! We'll call it a _distribution_ instead of a function. The Dirac distribution will be meaningless outside of an integral, but completely well behaved when in an integral."

Beautiful cycle between discovery and invention indeed!

[yor]

As a controls engineer, I've never considered a state-space equation in this fashion.
You blew my mind once again!

[zacharychristy]

"One application is relationships, the other is quantum mechanics, let's start with relationships"

Why did you go with the more difficult one first?

[ajbiffl]

"raise to the power of the derivative operator" YOU CAN"T JUST DROP BOMBS AND DIP

[alejrandom]

I learned a few days ago that
a -b
b a
is a matrix isomorphic to a+bi, it's nice to see that the exponent relation also holds :)

[josvanderspek]

We have to create a new contest besides Nobal, Fields, etc. This guy needs a medal.

[DebabrataMahapatra]

Incredible script man!
My best line: Cynically, this is abuse of notation; but charitably, this is the cycle of discovery and invention.
It's not just the visuals, but the script behind it is poetic. I love this guy!

[jamesjohns]

I teach graduate quantum mechanics to physical chemists, and I love showing them the videos you make, especially ones that visualize some of the maths relevant to QM. Please keep making them

[scrapmechanicgamer]

I love how in every video 3b1b goes from "that's utter nonsense" to "that's perfectly valid' and explains every step 0f the way

[paulhamacher]

I remember during my study where e to the power of a matrix confused the hell out of me. I desperately would have needed this video. Now I'm watching just out of pure interest. 😄