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

General exponentials, love, Schrödinger, and more.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/mat-exp-thanks
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
The Romeo-Juliet example is based on this essay by Steven Strogatz:
www.stevenstrogatz.com/essays/...
The book shown at the start is Vladimir Arnold's (excellent) textbook on ordinary differential equations.
amzn.to/3dtXSwj
Need a review of ordinary powers of e?
• What's so special abou...
Or of linear algebra?
• Linear transformations...
Timetable
0:00 - Definition
6:40 - Dynamics of love
13:17 - Linear systems
20:03 - General rotations
22:11 - Visualizing with flow
------------------
Code for this video:
github.com/3b1b/videos/blob/m...
These animations made using a custom python library, manim. See the FAQ comments here:
www.3blue1brown.com/faq#manim
github.com/3b1b/manim
You can find code for specific videos and projects here:
github.com/3b1b/videos/
Music by Vincent Rubinetti.
www.vincentrubinetti.com/
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
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Instagram: / 3blue1brown_animations
Patreon: / 3blue1brown
Facebook: / 3blue1brown

[@carl8703]

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

[@carl8703]

And yet this video still winds up convincing me, so if this is a joke, consider me fooled.

[@eyeborg3148]

Naa this stuff is all legit grant doesn’t mess around.

[@voliol8070]

@@carl8703 It could also work as a way to introduce/spread your own definitions, by ”fooling” the audience they are widely accepted. You’d have to be quite confident you can convice your audience though.

[@javidfarhan1675]

@@carl8703 I guess it takes a long time for him to make a video , and he wouldn't waste resources on an April fools prank.

[@collection6062]

EEEEEE EE EE EEEEE EE EE EEEEEE

[@freddyfozzyfilms2688]

When we needed him the most, he returned.

[@HarshGupta-zz9ur]

He Returned

[@oliverlong345]

What are the chances Freddy?

[@Robert_McGarry_Poems]

@@oliverlong345 Non zero...

[@Thalesfreitas96]

Return of the King

[@bothann]

My thoughts exactly.

[@Quotenbrtchen]

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

[@phucminhnguyenle250]

What a complex relationship

[@pralaykayal2476]

@@phucminhnguyenle250 😂😂😂

[@PMA65537]

love = 0 for all t is a solution (or I'm no tennis player).

[@NightmareCourtPictures]

"Hey e what's your favorite movie?" *e puts on sunglasses* the matrix

[@raymondchang8984]

@@PMA65537 Yo is this a detective conan reference?

[@maasbekooy901]

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

[@cnidariantide4207]

I've spent a half-hour on this video and I'm only one minute in-I'm cross-referencing everything he says. If you get on my level, you might enjoy yourself more!

[@wxcvbndu51]

"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

[@elias_xp95]

I'm here for fun, I'm not in uni and I'm not a mathematician. What's wrong with me!?

[@girlord13]

Your uni prefaced the content with the words 'fun little excercise'? That's just barefaced cheek!

[@carl8703]

At least they're not partial differential equations 🤷‍♂️

[@epauletshark3793]

@@elias_xp95 the same thing thats wrong with me, I guess you love learning and understanding things, and that this guy has a really nice voice.

[@jackychanmaths]

Ordinary differential equations are already very nice. Take a look at the partial differential equations......

[@zacharychristy8928]

"One application is relationships, the other is quantum mechanics, let's start with relationships" Why did you go with the more difficult one first?

[@ateium2409]

So that the second one would feel exponentially easier .

[@BudgiePanic]

beat me to it

[@ccooll2008]

I've got an exam on advanced methods in quantum mechanics and I can say: So far I got a good overview over quantum mechanics but Im still baffled by relationships.

[@7of475]

There's a reason I have been pursuing one of these things before the other, and it's certainly not that I'm proactive to the point of engaging with difficulty first.

[@YoutubeModeratorsSuckMyBalls]

Because nerds will never get a girlfriend) Just admit it, numbers will not help you with girls

[@onusbaum]

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

[@Tadesan]

What kind of money-work do you do?

[@dAvrilthebear]

Yes, absolutely!!!

[@fusionx4945]

Yes℅

[@yor1001]

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

[@andrewcihon-scott2170]

This video was great! I just started a graduate controls course, and this way of thinking about the state transition matrix really blew me away.

[@kelly4187]

@@andrewcihon-scott2170 I'm lecturing something called "Dynamical systems" for just one year and we've covered lots of this, characterising the fixed point of a 2D system of 1st order linear DEs by the behaviour around the fixed point of the system... And I'd never considered the relation to control theory. We need a grand unified theory of control dynamics!!!!!

[@micahrubel1356]

I legit just learned about state space in my first introductory course to control theory, and, while I probably don't appreciate it as richly as others, it still will probably help me with intuition going forward.

[@ajbiffl4695]

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

[@Kidynamo123]

Look up something called Exponential Shift

[@sosavoa]

MATH VIDEO WITH A CLIFFHANGER??? HUH

[@hybmnzz2658]

Well in his Linear Algebra playlist he did show how the derivative can be framed as a matrix because it is a linear operator. Remember in the Linear Algebra series how the derivative matrix acted on a polynomial? Now in this video he hinted at functions being like "infinite vectors". An offhand guess is that physicists describe functions by taking the coefficients of their power series into an "infinite vector". Put two and two together and I guess e^(d/dx) is a matrix which operates on functions. SPOILERS (?): According to wikipedia e^(d/dx) = 1 + D + D^2/2! + D^3/3! + .... where D is the derivative operator. Applying this to a function probably means taking the function, adding its first derivative, adding its second derivative over 2!, and so forth. Not sure if this is the same concept that can be reached by using what I guessed before the spoiler. Just speculating.

[@NXTangl]

@@hybmnzz2658 So, the MacLaurin polynomial centered on x, evaluated at (x + 1), if my mental derivation is correct...

[@edwardfanboy]

exp(d/dx) f(x) = f(x+1)

[@aaronbalan3959]

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

[@voxelltech]

same!

[@markgearing]

I don’t know what it is about the way he speaks, but it always sounds like he’s smiling. He’s the sort of teacher that any parent hopes their child will have.

[@KarelPletsStriker]

Every STEM student I know is in his debt

[@missmpolley9706]

That is so true!! I just thought about it and I can't think of anyone I know in an engeniering program that hasn't mentioned this channel

[@omniyambot9876]

You know Schrodinger right? Have you told him already?

[@kevinlee597]

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

[@rorysparshott4223]

God I wish I had it when I was one. I just shouted out loud asking why we weren't taught that a derivative of a vector can be expressed as a matrix multiplication. I wish I could explain how many months of second year that single thing would have saved me

[@JustanotherFormulaFAN]

Me watching this like 7 years before I need it

[@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!

[@shashankdevarmani]

Your professor is a genius 🤩. I just love those aha moments when something just clicks and which later on give you so many moments to grin while solving examples 😁

[@jamesblackburn8110]

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.

[@vigilantcosmicpenguin8721]

It's always nice to feel like Calvin every once in a while.

[@uncreativecosmos]

Who doesn't love Calvin & Hobbes ? Lovely analogy by the way.

[@sarahmargaret234]

I made $ 6,500 profit just within 14 days of my Investment with an initial amount of $ 2000 all with the help of Mr James

[@shelderevanston5864]

Yes I'm a living testimony of Mr James trading services , he has really changed my financial status for the best

[@johncluff5986]

@Mary Jane how do i start trading with Mr James

[@joaofrancisco8864]

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.

[@pechabada]

You are right. I can only regret that such a thing did not exist 40 years ago, when I needed it. Now all I have to do is admire the beauty, simplicity and clarity of this way of interpreting mathematics.

[@volodymyrhavrylov7993]

+100500 man, ur so right

[@easymathematik]

It is freely available, because it is profitable. Such videos takes a loooooot of time. Nobody would do it for free. 20 years ago I am sure he would try to make a VHS or DVD series. But there is no guarantee for selling. KZbin is a guarantee. Patreon is a guarantee. If he would not get paid he would not do it. Money is everything. And as long as money is "enough", such videos will exist.

[@RolandOrre]

but it's not always trivial to download it, so you really have it, I first failed, but then found that I had to update the downloader software, as google video of some reason change their protocols so often.

[@alejrandom6592]

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 :)

[@JosiahWarren]

Never thought of that. Thank you

[@angeldude101]

A topic I've recently been learning about is Clifford Algebra, and while it usually isn't done with matrices, the various objects have matrix representations. The algebra also encodes the complex numbers. That matrix is the actual matrix representation of a+bi in Clifford Algebra, despite the fact that the the unit vectors it usually acts on look completely different in matrix form than normal unit vectors.

[@agrajyadav2951]

Ja

[@adjoint_functor]

This actually looks shockingly similar to the inverse matrix formula, wonder if there’s a connection there

[@PersonaRandomNumbers]

@@adjoint_functor If you squint, the formula for the inverse of a 2x2 matrix kinda looks like the formula for the multiplicative inverse of a complex number, z^-1 = z*/|z|; that is, the complex conjugate (the number but with the imaginary part multiplied by -1), divided by the magnitude of the complex number. If you also watched 3b1b's videos on complex numbers, the relationships are kinda neat. You can see from that formula that the magnitude of a complex number is equal to the determinant of its matrix representation, and the multiplicative inverse of a complex number multiplying to equal 1 is mirrored by the matrix representation's inverse multiplying to the identity matrix.

[@josvanderspek1403]

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

[@integralboi2900]

The king has returned. Let's hope this isn't the start of a new hiatus!

[@MaxC_1]

He was busy with courses in some uni, think MIT and prolly something related so makes sense. Also maybe planning and stuff too

[@fzigunov]

I would estimate >200 hours of work to get this video done. It would actually be nice to know how much time 3B1B actually spends on these creations. We're blessed and don't even know...

[@jamienorth1603]

@@fzigunov ive used manim (the language he used to make his animations) a little bit and it honestly takes about an hour to make one scene so god knows how long it takes him. Propts to him for this

[@LimLux]

@@fzigunov He has a team for stuff like that?.

[@fzigunov]

@@LimLux I think you would be surprised...

[@scottarmstrong9566]

As a math professor teaching ODE this semester, let me just say: this is wonderful!

[@samsonblack]

Same. I recommend both the Diff Eq playlist and the Linear Algebra playlist to my students every term.

[@Jackisaboss1208]

@@samsonblack I recommend the linear algebra series to my classmates every term 😂

[@sauronstillgood6804]

@@Jackisaboss1208 how tf you take linear algebra every term 😂

[@zacharyeichenberger4929]

@@sauronstillgood6804 CE, probably

[@scrapmechanicgamer3155]

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

[@paulhamacher773]

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. 😄

[@mathemaniac]

It's been a long time since you uploaded. Welcome back!

[@3blue1brown]

Thanks! I love your symmetries video, by the way.

[@mathemaniac]

@@3blue1brown Wow thanks so much! Never thought you would reply!

[@jamesjohns2201]

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

[@arhanvora4545]

Heyo, would you mind telling us which videos you have found particularly relevant to QM? I’d really like to dive into them. Cheers :D

[@pvlkmrv]

"Why would mathematicians and physicists be interested in torturing their poor matrices this way?"

[@PMA65537]

Wait till you hear what Hilbert did to lettuces.

[@bradenhuss7003]

"torturing the MATRICES?!?" -college undergrads after spending two weeks straight calculating the inverses of 3x3 matrices by hand

[@GeorgeTsiros]

Sweet, sweet revenge.

[@bluelemon243]

@@PMA65537 can you tell more ?

[@PabloNL89]

Incredible. Incomparable work. As a mathematician, I can give you credit that maybe a regular suscriptor can't. I swear your channel have amazed me and you deserve some kind of great well paid recognition.

[@DebabrataMahapatra91]

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!

[@stevenlin5377]

I just want to say that you're my favorite youtuber. You help me so much during my study in university and getting a master degree. You also reignite my interest in math, which I previously thought wasn't even possible. Words can't express how much I love your video and the way you teach math.

[@3blue1brown]

That's so kind of you, thanks. I hope you help to spread the love of math with others in your life :)

[@domenicobianchi8]

@@3blue1brown I came searching for the sequel. I was sure i watched it years ago, but now i realize maybe you never published it. Am i right? (By the way, terrific job. Im a long way fan of the channel, i'm currenly ill, and binge watching all your videos-again!)

[@billiemelo4276]

I’m highly appreciative that the matrices at 5:14 are the first 9 digits of pi lol. Great video as always, thank you Grant!!

[@niteman555]

I wish I had the necessary math background when I was taking advanced physics in college. it would have made the course a lot less painful.

[@LoptukqrickL11]

haha right? At the end I was like "oh, we've basically been doing the same thing over and over with slight variations."

[@Redstonmaster]

Mathematicians before e to the x extends the definitions: "We do some more theorems and we finish with math" Mathematicians after:"What if we exponentiate EVERYTHING"

[@SimonClarkstone]

Associative algebra: *exists* Mathematicians: "Time to define e^x on it"

[@nolan412]

They're paid to answer these questions because students ask them.

[@jakubszczesnowicz3201]

@@nolan412 They're paid to answer these questions because there's a chance someone in the future will use their work and apply it to the real world

[@durnsidh6483]

Can't you define e^x on arbitrary rings?

[@andremouss2536]

@@durnsidh6483 Yes, if the ring has a metric

[@adityachk2002]

I was like it’s been 3 months since his last upload and 6 months since his regular uploads

[@3blue1brown]

Things will pick up again here now. The videos might not come _super_ quickly, but there are a lot of things I'm excited about coming up this year.

[@tanmaydeshpande]

@@3blue1brown excited for your new stuff. Loved all of your other content! (Btw, have you ever considered becoming a voice for asmr videos?)

[@adityachk2002]

@@3blue1brown 🤩

[@GolumHD]

@@3blue1brown including blockchains? ^^

[@kathanshah8305]

@@3blue1brown yesterday I had commented on your previous community post and today here you are

[@lancediano8014]

Nothing but respect for these videos, teaching math to people from all levels of base knowledge is already hard. The fact you manage to teach them while giving flawless animations and graphical representations is beyond me. How you actually do these animations must take forever, watching them and seeing the numbers count up when u were multiplying matrices had to take a while to figure out and I respect the work you put into it. Lastly this was my first time watching ur videos on my new oled tv, fantastic experience!

[@salimchahine8324]

without a doubt one of the best math channels on youtube, if not THE best! waiting for episode 7 btw.

[@StevenHodder]

An entire semester's worth of my electromagnetics class in university just made sense after 20 years...

[@Hansengineering]

There are many such classes where that happened more or less as I was taking them because of this channel.

[@ultimategotea]

EM is the hardest part of physics to grasp

[@chalichaligha3234]

@@ultimategotea , quantum electrodynamics enters the chat.

[@ultimategotea]

@@chalichaligha3234 that's not physics, that's witchcraft

[@azpcox]

The summation of the wave functions on Schroedingers equation makes much more sense now than it did in college. Grant, where were you 30 years ago? ;) Thanks for making math exciting again! Want more STEM excitement? Show these videos in junior high and high school. Maybe earlier.

[@metaljoe9923]

"Often this is a function. (But whatever, functions are really just infinite-dimensional vectors)" I hope that parenthetical is a teaser for a future video on Functional Analysis.

[@sudeepgopal]

I did my engineering sometime ago, I remember seeing the e ^ (matrix) and I remember losing it and thinking wtf....You really have a way of explaining things so elegantly.. I hope I encounter these kind of problems in my career - Very enlightening....Thank you!

[@SashaTownsendTulsa]

Thank you so much, Grant. That was beautiful! Our college cut this chapter from our elementary differential equations course, but I’m still going to share this video with my students!

[@ayushdeep7900]

He came back when we missed him the most because that's what heroes do.

[@purungo]

guess my father's not a hero then

[@ayushdeep7900]

@@purungo you shouldn't have written this

[@purungo]

@@ayushdeep7900 Yes I should

[@ayushdeep7900]

@@purungo sorry bro

[@HebaruSan]

So relieved when the Romeo 'n' Juliet graph didn't trace out a heart shape

[@billowen3285]

Why?

[@pierrekilgoretrout3143]

So disappointed that Romeo 'n' Juliet graph didn't trace out a heart shape :'(

[@diekleinehexe3620]

But how would those differential equations look? 🤔

[@drjohnsonnn]

Dy/Dx = heart

[@santerisatama5409]

@@pierrekilgoretrout3143 Don't feel bad! Here, have a vertical ice cream cone/pointy hat of an elf seen from behind:

[@OwenMcKinley]

Grant, you are one of the most important educators in the modern world. To have your library of presentations in my pocket at all times is one of the best feelings there is. Math went from my least to to my most favored topic of study because of your content. May you continue to be an ambassador in this field!

[@sholinwright6621]

For me this came up in state space methods in control theory. We broke a complex higher order differential equation control problem into a set of first order differential equations, collected them into a vector, and solved the matrix exponential. The advantage was that we could completely assign the poles (roots) of the controlled dynamics of the system with matrix feedback.

[@jalexandus]

What I love about your videos is that you might not understand parts of them at first but as you go back to them later, perhaps after gaining some knowledge, things start to click.

[@maxsch.6555]

I was waiting two years for this video! (the preview was shown in the first video about DE) Im so happy right know! Thank you! 😍

[@3blue1brown]

I don't always make the follow-ups I promise quickly (*ahem* probabilities of probabilities *ahem*), but for those willing to wait a few years I feel comfortable promising all will come in due time.

[@marvelritik8937]

@@3blue1brown yeah I've been waiting for possibilities of possibilities 😆😆

[@thecoloroctet1365]

I was so sad cuz I thought the DE series was discontinued, but now I’m hyped and can’t wait for the rest it!

[@King-Matshobane]

Thank you for the time and effort you took to explain these concepts. I intuitively applied them in my classes without a clear understanding of what these numbers are doing. Visualisation at a level like in your presentation is more like a peep show on what’s going on in the CPU. Finite Element Analysis and Computational Fluid Dynamics applies these concepts. Once again thank for the good work you are doing. 👏🏾👏🏾👏🏾

[@ben_car_8115]

I just have to say, your videos are consistently amazing. I am taking a quantum mechanics class right now and needed to brush up on my linear algebra since it’s been a few years, I automatically knew I could watch your videos to regain an intuition and it worked. I have a much better intuition for some of the operations and ideas than others in my class because of these videos. Specifically with matrix functions, these were literally just brought up and while it looked kind of odd, my teacher mentioned he would give us some easy ways to evaluate these. I didn’t even know about this video, but now I have a visual intuition for this, in the context of quantum mechanics! 10/10, you are a god among men.

[@alxkeda]

Happy April fools!

[@bhavyavashisht5891]

First

[@3blue1brown]

The joke is that I actually upload content these days :)

[@Offthedog]

@@3blue1brown joke or not, I’m still grateful that you still upload content. Your channel is a great contributor to making me pursue a career in applied mathematics.

[@jockycracker8253]

​@@3blue1brown Then maybe an "essence of probability" as a joke, hmm?

[@yoyokojo651]

@@3blue1brown i was going to like this comment till i saw it that it had 314 likes, oh how nice things can be

[@RJC65__]

Im literally learning this in class rn and im like, oh well ig 3blue1brown will be my teacher today

[@taymorrison]

this is what i love about 3blue1brown, i have to carry out these calculations in my qm class all the time, but i have absolutely never understood the intuition behind it until now

[@nightlord531]

Really hope you explore Laplace transforms later in this series! Using them to solve DEs makes designing control systems much easier.

[@rv706]

This video was so lovely. A motivating thing that you could've added, together with the fact that you already said that matrix exponentiation solves linear dynamical systems, is that the latter are the first order approximations of non-linear dynamical systems around equilibrium points. You would actually have enough material on this to make a whole video, explaining linearization and e.g. normal modes/frequencies of (small) oscillations.

[@brahadkokad5424]

Finally!!! I really missed those pi-creatures (and also your voice, Grant! It is hypnotizing enough to make people stick to your videos without giving up, no matter how fighting the math in them can get sometimes😳)

[@anywallsocket]

Seeing the pi’s always miss each other with their heart eyes is honestly tragic 😍😭

[@PianoPsych]

I recall with her great fondness my first contact with this subject matter in an early edition of Hirsch and Smale’s Differential Equations, Dynamical Systems and Linear Algebra. Takes me back many years. You’re extraordinarily good at this. Truly a treasure. Thank you for these videos.

[@DeltaSleepy]

Your illustrations and explanations are second to none. Great content!

[@arda8393]

This is the only channel where im practically lost from the get go but still compelled to watch till the end

[@jimbobur]

Oh my god I wish I'd known this years ago. They just threw this at us in QM lectures during my physics degree and not one lecturer ever bothered to explain any of this, even briefly..

[@theyredistortingyourrhythm130]

But, are you aware in 2021?

[@UmarO]

I was gonna make this comment

[@DD-vc7fq]

I am getting a masters degree in mathematics in 3 months and yet, I have never seen matrix being an exponent. lmao.

[@Shadowkainine]

This video hits different after having had my first proper differential equations course, it all feels so familiar now. Please keep these coming! They are so much fun!

[@rommelchinas4593]

This is certainly one of my 3B1B favorites. Thanks for making these videos.

[@MrPointness]

6:35 I was fully expecting for it to go like "let's start with the simpler of the two, quantum mechanics"

[@tomassanchezpucheta2142]

This channel is one of the greatest things that has ever happended to the math explanation!

[@julesthomas3335]

20:14 "It just seems like your feelings aren't real" 😂 In fact I really love complex numbers

[@d3scripted672]

i'm only in elementary school but i love this channel. It really just shows the beauty of math, and inspires me alot to learn more. Thanks for making these high quality videos, much love and respect (:

[@a00954926]

This is hands down one of the best channels ever

[@maxwellsequation4887]

The*

[@ultimatedude5686]

@@maxwellsequation4887 They wrote it correctly

[@farrankhawaja9856]

Grant is the kind of guy that can combine love and the Schrodinger equations all while raising e to the derivative operator! Btw I just wanna say that ever since you've 'disappeared' I started to slowly drift away from math. Thanks for inspiring me again! Can't wait for ur new uploads!

[@ishaqkhan1024]

Thank you, I can always rely on this channel for a comprehensive explanation of these topics which seem very complicated and daunting at first. There is clearly a lot of effort put into these videos, and they are very much appreciated.

[@vyaasgururajan932]

This is beautiful. This is mostly why Dirac notation is so powerful: Dirac must have similarly reasoned that Schrodinger's Wave mechanics and Heisenberg's Matrix mechanics were essentially describing the same thing!

[@JoBrew32]

I remember the absolute outrage of my class last semester when we saw this for the first time. Several people called it abuse. This vid makes it seem much nicer.

[@jaredellison326]

What class was it?

[@valerianmp]

Probably intro to diff eq

[@SimonClarkstone]

Works on any associative algebra, AFAICT.

[@hybmnzz2658]

Back in the good old days inventing math was about abusing notation until it wasn't abuse anymore

[@MatthijsvanDuin]

@@SimonClarkstone well no you need some more conditions, it needs to be an algebra over ℚ to allow multiplication by 1/n! and (unless you're satisfied with only being able to use it on nilpotent elements) you'll need a topological structure that defines convergence.

[@ajhcornwall]

I definitely want to hear about e^d/dx, and more on Schrödinger's equation. Brilliant!

[@davidkigel3861]

Thanks for another amazing video @3Blue1Brown! I think it could be also worth mentioning the alternative definition of the exponential of a matrix as the limit of the products (1+A/n)^n when n tends to infinity. What I like of it is: - Immediately visualizable for a given n as a simple Euler numeric integration in n steps. - Can be motivated and bring intuition from simple problems with regular numbers as continuous compound interest.

[@thelonegerman2314]

I'm Interested in the DIAGAGNOLIZATION of matrix VECTOR using Taylor expansion and DETERMINANT for infinite series and PAULI MATRIX and Clifford Algebra for solving the Schoringer Equation And HEINGBER uncertainty for QUANTUM FIELD PERBUTATION

[@finojose]

Grant, This video is awsome!. I would like to thank you for all your videos and helping me to progress in my development in quantum computing and machine learning. Your visual, intuitive and elegant way to teach linear algebra and related concepts helped me a lot to move ahead and get in love with the field. Thank you very much.

[@ayadevin2413]

The timing could not be more perfect - just went over this topic in class and 1 week before the midterm!

[@CharlesPanigeo]

Gotta love power series. They make it possible to define all sorts of functions on matrixies like cosine, sine, exponentials, logarithms, etc.

[@PileOfEmptyTapes]

I haven't dealt with any of this in over a decade, but what an absolutely beautiful presentation! Hats off to you, sir.

[@payamism]

The kids today have so many excellent sources available to them in a couple of clicks. Back in the day, even 10 years ago, we didn't have such clear, attractive, and deep explanations. Most people, even the ones who would get the A+ really didn't understand the concepts. They were just good at taking the test. I wish this was available to me 20 years ago when I was a freshman in college.

[@seahyx120]

Now everyone, not just the best scientists and researchers out there, can stand on the shoulders of giants. This is what the internet is made for.

[@songhetang2961]

I am so excited to learn maths this way from you. I have a PhD in physics, and am more than 20 years into my career as a scientist. I do a lot of calculations to support my experiments, so I have a basic understanding of mathematics, but I feel like a fog is being wiped clean when I watch your videos. Thank you so much!

[@sohan239]

Notice the pi's eyes and expressions as the graph changes!! Incredible attention to detail grant! Amazing.. 10:00

[@mossmanmick728]

I remember receiving a question on my linear algebra assignment this year involving this method except with 4 different variables. If you can imagine it took a while (several hours casually doing it) as we had to do all the workings by hand! But it was possibly one of the most memorable questions I've ever done

[@diekleinehexe3620]

I'm a third year physics student and by now, I have seen a fair bit of QM. It's amazing how you always manage to give new and visually intuitive ways of imagining calculations, that we do so often without further thinking about it. I wish our lectures would be taught more like this.

[@gena8414]

leave physics. you are wasting your life.

[@michaelking8391]

@@gena8414 what isn't then wasting your life?

[@gena8414]

@@michaelking8391 cs

[@velhacega]

@@gena8414 so you can be dumber than a physicist that can code just as well as you?

[@gena8414]

@@velhacega lmaoooooooo show me a physicist who can code as well as me

[@gabedarrett1301]

This is indescribably beautiful! You have a gift, Grant!

[@davidbenz2280]

Grant. In the "visualizing the flow" section, you forgot to mention that when you "attach the vector Mv to v", you are scaling Mv down, and indicating its magnitude with color. I had seen you use that technique before, in one of your diff Eq videos, and I eventually remembered it, but in this one I kept asking myself "What does 'attach the vector Mv' mean?" And "where does the "new", reduced magnitude of Mv come from?" The direction of Mv remains the same, but in your visual, Mv "shrinks" down as it slides to the tip of the arrow of v. This is a direct result of artificial scaling, to remove clutter. You are a master mathematician, and computer illustrator/coder. I will never wander as skillfully through the fields of mathematics as you do, but I appreciate so much that you have left a few bread crumbs along the way for me to follow. My degree is in the physical sciences from Berkeley in 1983. I was forced to minimize my math curriculum, in order to preserve my GPA. But your videos have enabled me to go back and learn more about what I couldn't comprehend then. I am so very grateful to you, for your stellar efforts to enhance our understanding of calculus, linear algebra, and differential equations. I feel like I've come full circle, like exponentiating a rotation matrix, or an imaginary number!

[@cr4zyg3n36]

I wish you had been my math teacher... you do the world a great deal of justice with these wonderfully dynamic animations. Thank you.

[@deepakramalingam6041]

Never understand it fully but came for the channel

[@iremsipahi4507]

Literally me

[@U20E0]

If you don’t understand a math related thing, go see if 3blue1brown has a video on it, it might help yo- *wait a second...*

[@giabao576]

as someone who studies physics with the "just enough" math knowledge, this video is mind blowing to me, it makes me realize that I should be looking more into the math of the physics phenomenons. Physics is applied mathematics, but physics also spark ideas, problems for us to develop models, concepts that we have never thought about!

[@bunyaminozkaya498]

I don't normally comment on videos but I'm SO GLAD you continue this series !!!!! I'm happy like a child :)

[@Arthur-xe3pu]

Currently learning about eigen problems and their applications in physics, it's nice to see that you'v come back to this series, great video.

[@thelonegerman2314]

I'm Interested in comparison of matrix VECTOR , EIGENVECTORS using Taylor expansion and DETERMINANT for infinite series and PAULI MATRIX

[@SgtSplatts]

Girlfriend: "I want to move in" Me: busts out a matrix

[@vigilantcosmicpenguin8721]

Well, that'll just introduce another variable of what it's like to live together. Better get ready for some really complicated calculations.

[@aela7060]

Scrolled for this

[@AdarshSingh-wv4ff]

I like the pun @20:16 where Juliet says "it just seems like your feelings aren't REAL" When you introduced the Complex version of the solution... I just calculated @26:35 & found The exp{ d/dt } is (in a sense) an Upper Triangular Matrix compressing the whole Pascal's Triangle... Now I'm looking for its application if any. I wonder where did you find this example??

[@codyfan7161]

-ihbar d/dx is the momentum operator in QM

[@Himineejimineee]

but in the expanded form you’ll need to matriculate the real part of the operator, so it works out to be more of a square-like version of Pascal’s triangle. We use it in fluid mechanics for dilations of compressible flow 🤡

[@s.miladm.mousavi4975]

Thanks for the great video, really inspiring and satisfying, still waiting for next chapters!

[@alexvaltatzis9139]

Your videos are the best mathematics video I've seen. So helpful. So inspiring. Such good work! Keep it up!

[@OuroborosVengeance]

I thank dear Zeus and the olimpians for bringing you back. You made my week with this video. Thank you 3b1b! May your week be good too!

[@Mutual_Information]

There is only one KZbin channel where I hit the like button before I even watch it.

[@definesigint2823]

I've often wondered...can creators see _when_ you upvote (possibly giving some data on the moment the video was meaningful)?

[@Mutual_Information]

@@definesigint2823 Turns out I’m a creator :) on KZbin studio, which shows analytics, no you can’t see when it happened. However, you do see percent of viewers who are still watching over the duration of the vid. So if that percentage drops, you know that part of the vid gets people to leave. Pretty useful.

[@guest_of_randomness]

@@Mutual_Information lmao i m sure the romeo and juliet part hold the people in this video

[@jn5433]

Why the fck would you do that.

[@Mutual_Information]

@@jn5433 lol

[@christmassnow3465]

I never attended math courses beyond those of high-school, for reasons well beyond my control. I eventually learned from some textbooks which I bought on my own. Although I never knew about the matrix as an exponent, the highlights of the solution were quite clear. The bits of information fit well and give a good picture of the line of thought.

[@Aryan-hn6ei]

wow what a nice visualisation and the way you let me understand th concepts is completely nice . i think that you should publish a book related to these mathematical modelling .

[@nikoovarela5723]

Never have i clicked so fast on a notification

[@hashirhussain.]

Even me. I was watching something else. I ran to see this .

[@maxsch.6555]

Same

[@apocalypt0723]

me 2 :D

[@nullpointer1755]

Same

[@skilz8098]

@@hashirhussain. There is one other KZbinr that has this same effect and that is Ben Eater!

[@nk9083]

Great contrast on the Textbook Progression vs. Discovery Progression. I wish more lectures and textbooks would start with the discovery progression. I remember way back in High school calculus - It is especially dangerous when a teacher who doesn’t understand the subject blindly follows the textbook progression.

[@estebanguerrero682]

WHAT AN ABSOLUTE MASTERPIECE!!! You make an inspiration to keeping up the studies

[@dennisbrown5313]

Absolutely one of the most interesting, clear and useful "advanced mathematical" concepts anyone has presented. As someone who has done D.E., all the calculus', and even a first level quantum course (using normal wave equations) I have always wondered how matrix's are used in the Schrödinger equation! I am going to reteach myself QM using matrix approach (after getting up to speed on tensors - ability to be base's independent is critical to be comfortable with Dirac bra-ket notation. I also want to see how Geometric Algebra (Clifford Algebra) simplifies a lot of this.)

[@ornessarhithfaeron3576]

13:00 reminded me of my robotics class! It's one of the matrices we use to describe the rotation of robotic arms :)

[@georgerodionov5941]

"Rotation in a kind of function space". Thanks, my brain is now all over the wall

[@jonathanmartin2326]

It seems 4 other people laughed out loud at this comment.

[@MatteoMori]

My room has a new gory wallpaper as well

[@davidwigginton8328]

I love your work and I love this channel. I want more on this series, can’t get enough!

[@onusbaum]

This is BRILLANT, it explains complex numbers intuitively !

[@Tehom1]

Even stranger things have been exponentiated to. Strangest one I know: the creation and annihilation operators. It comes up in the analysis of squeezed light, such as LIGO uses. It's convenient to analyze squeezed light by changing the basis, by a Bogulyobov transform, to the sinh and cosh functions, where the arguments are the annihilation operator plus or minus the creation operator. Sinh and cosh of course are just e^x+/- e^-x. So they were exponentiating to the power of "annihilating a photon minus creating one" etc, and it actually works to describe a physical process.