Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

[@Thien--Nguyen]

We are probably the first generation ever to witness eigenvectors and eigenvalues and linear transformation animated, in motion as nicely and as accurately as this. We are very lucky to be in a time of incredible technologies and incredible people like 3B1B Grant here. Thank you!

[@rosszhu1660]

Yes, indeed. You know during my time in university, nothing could help except your imagination, and you must verify any single piece of hypothesis with pen and paper.

[@don9526]

Yes your generation is very spoiled.

[@klam77]

True! When i went to school.....NEVER explained like this!

[@vannilesoep]

@@don9526 This generation isn't spoiled, it just has the unique opportunity to gain 𝘪𝘯𝘴𝘪𝘨𝘩𝘵 in these mathematical topics, instead of just learning a trick. And this insight could enable a much deeper understanding of mathematics or other topics, which could in turn spark new ideas or gaining more knowledge. Your comment implies a somewhat negative attitude towards this generation, we could also flip this reasoning around, saying "Your generation consisted of learning monkeys new tricks, the current generation consists of reaching insight and understanding", but lets not do that :)

[@rayzoldyck8044]

@@don9526 OK, boomer.

[@madelinescyphers5413]

After having a degree in math, and working on my master's in optimzation, after audibly went "ohhh" when he first explained what eigen vectors and values were. Like, I finally get it. It's more than just some abstract thing that I need and use. These videos are golden.

[@3blue1brown]

Thanks so much!

[@Milark]

@@3blue1brown changing lives dude!

[@wesm6747]

I just got my degree in Computer Engineering, and I'm working on my masters in Computer Engineering. Same thing here. I finally get it :,) These vids are amazing

[@yaacheese8643]

@@wesm6747 Where do you use Eigenvectors and Eigenvalues in Computer Engineering if you don't mind me asking?

[@wesm6747]

@@yaacheese8643 I've used it in a few Electrical Engineering classes. They've been more prominent in a scientific computing class I took in grad school. I think they also come up in comp graphics.

[@Cena199205]

I cannot thank you enough for this awesome series. Like others, I have a master degree and I still don't fully understand some of these basic concepts! Even after 6 years of publishing this series, it is still the best series explaining linear algebra.

[@RealEngineering]

I wish I had this in college. I struggled with this subject so much

[@3blue1brown]

Thanks! Hopefully, current college students find it helpful. By the way, just watched your transistor video and loved it!

[@RealEngineering]

They definitely will and thank you!

[@pabilbadoespecial]

Hey, glad to also see you here, love your videos!!!!

[@JohnCLiberte]

CGP grey should also visit. Actually this 3b1b voice sounds like greys ..

[@bryanjordan1263]

I follow both your channels religiously (I'm an electrical engineering/neuroscience student in Sydney) and just floating the suggestion that if you two did a 3Blue1Brown X Real Engineering series exploring the physics involved in aeronautical/aerospace applications (wouldn't hurt looking into other engineering domains (eg. electrical lol) and examining other spaces of mathematics such as complex numbers) - you would be true MVPs of KZbin/academia (pretty sure they're considered equivalent).

[@thenumber1723]

This person is the single most influential, and the only person around, in my life who made me understand the concept of Eigenvalues and Eigenvectors and their essence. God bless people like Grant who made themselves available (through online channels) to individuals who don't have such teachers, with positive influence, in their life to explain such complicated topics with fine clarity and simplicity :)

[@Alvin-yi3il]

Definitely

[@oraange]

fax

[@michaelkotthaus7120]

This is really good explained and the animations are delightful. For the viewers without any knowledge of the German language, it may be interesting, that "eigen" can be translated to "own" or "itself". So, an eigenvector is an "itself-vector".

[@zinzin7075]

Thanks!

[@drizer4real]

Same in Dutch

[@pedrolinscosta]

In brazilian portuguese, we call them "autovalores e autovetores", which would sth like own-values and own-vectors, respectively... 😊

[@wiiznokes2237]

In french is "vecteur propre"

[@TheHitchhiker]

Of course english is the only language that leaves it in german.

[@chadmaster6936]

"I wont teach you how to compute them" - Proceeds to teach us how to compute them better than any textbook or professor ever could

[@gracialonignasiver6302]

His explanation on why it's computed the way it is completely blew my mind. For nearly 4 years now I could compute Eigenvectors and I never understood why I was doing what I was doing. I seriously had to pause his video, get up out of my chair and pace around my room to let it sink in. Absolutely amazing.

[@chadmaster6936]

@@gracialonignasiver6302 I only ever heard of eigenvectors.. never learned them (I was in hs when I first watched this) and I did the same thing where I stood up and was like "what did I just watch"

[@paulbarton4395]

It can get pretty involved, his example was a 2 x 2 'upper triangular' matrix, which is why it turned out nice...but a 6 x 6 matrix thats not upper triangular will require some work, like Gaussian elimination, followed by finding the zeros of a 6th degree polynomial. Thats what computers are for tho

[@erilgaz]

@@paulbarton4395 but a 6 x 6 matrix that's not upper triangular will require some work, like painstakingly typing 36 numbers into wolframalpha and pressing enter.

[@Manik481]

@@gracialonignasiver6302 same here :)

[@tomtom9509]

I understood more about Eigenvalues and Eigenvectors in 15 min. than I did in two years of math undergraduate course. Thanks a lot. and great animation work too! It was the same for derivatives and integrals. I did great marks in high school in physics and maths but I truly didn't get why derivatives and integrals were working for physics. For me it was magic. I learned the formula and applied them, but it was just black box techniques. It is only at university that a friend of mine in 10 min. explained their meaning to me and everything became crystal clear. Those 10 min. simply changed my life. I think teachers should be every attentive to this.Take some time to teach the meaning, the big picture and only then get into the nitty gritty details.

[@kevinbyrne4538]

There's a saying: "Those who can, do ; those who can't, teach." However, teaching is also an art and a skill. It was often a shock to me at university that accomplished scientists were often bad teachers.

[@StefSubZero270]

You are right! I just completely finished my geometry (linear algebra) course at Physics department and i have a tonshit of doubts about it and i have the exam in 1 months. I'm struggling do study it and solving exercises (because i also have other courses i have to study to obviously), but finding these channel helped me A LOT to understand what was my professor talking about :D

[@CombraStudios]

15 min? are you watching math on 1.25 speed?

[@mohab.m.metwally]

sometimes i do in 2X depending on motivation, attention, professor age :)

[@niemandniemand2178]

dumbass

[@calexito9448]

As an INTP, I can't be thankful enough about this awesome content. In maths, I struggle with the actual calculations and the formulas but I recently discovered the intuitive way of learning maths. This series exactly targets that. After understanding the concepts I'm able to deduce the actual formulas and properties without having to rely on memory. This is the very first donation I do in the entire internet and I couldn't think of anyone better than you. Thank you so much.

[@Robert-jy9jm]

What kind of monster would downvote this masterpiece? This may very well be one of the best series ever made.

[@MrCmon113]

Robert What are you talking about? The votes are overwhelmingly positive.

[@alexander-jl6cs]

Ssshh! It's people from Australia and New Zealand.

[@Chrysaries]

@@alexander-jl6cs Oh yeah, their votes scale with an eigenvalue of -1

[@laurv8370]

indeed one must be a complete moron to downvote this video... I bet some frustrated math teachers are in that list (former math teacher myself)

[@steves1015]

Probably those who say “i hate maths”... ;)

[@ProfessorEisenoxid]

I cant name one video producer who has such an enormous positive feedback and with viewers who are so fascinated by the content!

[@3blue1brown]

You commenters are the freaking best. Usually, KZbin comments can be such a dark hole, but every video I've been uplifted and pumped to make more.

[@ProfessorEisenoxid]

+3Blue1Brown Therefore, not only your explanations are higly intuitive yout animations fit and are beautiful, a fine piece of video-art! I am looking forward to every video!

[@bennick7568]

I hit a wall studying Eigenvalues and vectors in college and decided I had had enough math. Learning about eigenvalues and vectors here, in such a clear and enjoyable way, gives me confidence and makes me want to keep learning. Thank you!

[@VainCape]

you deserve the nobel prize in maths for making math accessible like this to millions of students

[@douglasespindola5185]

Since that there is no math nobel prize, a Fields Medal should do the work. And yes, Mr. Grant deserves it!

[@michaellai5549]

Totally agree

[@faizanpathan8645]

hi, I still have a doubt at 10:34 it shows some non zero vector when squishes to one dimension it becomes 0 . But I have a doubt that it should be reversed according to matrix multiplication that in one dimension we have to find some non zero vector that becomes 0 according to our first basis vector that is [1,0] and [0,1]

[@nalat1suket4nk0]

there isn't a nobel prize in math

[@beachboysandrew]

@@nalat1suket4nk0 if there were, we all know it would be mostly won by Israelis XD

[@Emoechaiti]

0:16 The beauty of music lies on how we perceive it (decoding process of sound in our brains). But the beauty of mathematics, even though everyone has an inbuilt intuition about it just like music, still people don't understand because they can't relate the numbers, symbols, methods, formulas, graphs, and other mathematical entities with the reality (existence). While Mathematics is all about reality. How frustrated would someone be if they can't relate the written musical notes with their respective sounds !!! The way you teach is honestly the best way to understand mathematics. Your hardwork in the field of your interest is clearly visible in the beauty of your teaching. Thank you sir 🙏 And keep inspiring us

[@dixit6605]

Can we think of a single mathematical concept that does not relate to reality? All of math seems to come from how we have defined counting (we count things, space, time, etc.) and counting is a reality. Once counting has been defined, processes of applications can logically be defined on them like operations, relations, functions and many things. But is there any single thing that is not based upon reality, i.e counting? Perhaps, mathematics is indeed the manipultion of numbers (i.e, the tools for counting).

[@zhuolovesmath7483]

It's so lucky that this series is already complete when I'm studying linear algebra

[@faizanpathan8645]

hi, I still have a doubt at 10:34 it shows some non zero vector when squishes to one dimension it becomes 0 . But I have a doubt that it should be reversed according to matrix multiplication that in one dimension we have to find some non zero vector that becomes 0 according to our first basis vector that is [1,0] and [0,1]

[@jhonnystiven]

​@@faizanpathan8645to find that vector you don't do it with an inverse matrix since matrices with a determinant of zero cannot be inversed. This series of videos explains this in the chapter about linear systems of equations. Because our vector equals to zero when multiplied by our matrix, what we need to find lies in the null space, which you can find using row echelon form and solving the linear system of equations from there. Look for explanations on calculating the null space on videos from other channels, like Khan academy.

[@floatoss]

​@@faizanpathan8645could you be more specific? Grant is basically saying that, if you calculate "what vectors, when pumped through this matrix (A - {lambda}*I), land to the zero vector", they are essentially your Eigen Vectors for the matrix 'transform 'A'. You are needed to calculate "what vectors are going to land to the zero vector when you pump through that matrix".

[@faizanpathan8645]

@@floatoss thnx , now I got it

[@notrhythm]

yeah

[@swinfwar]

Every student in introductory quantum mechanics needs to watch this video. These concepts are extremely important to QM and it really clears up the importance of the eigenstates of operators. Thank you for the great explanation!

[@biznock09]

It's amazing. I fell in love with linear algebra because of its computational power and knew there was intuition buried in the numbers. I frequently, if not always had my questioned that I could only express at the time using "visual vocabulary" ignored or interpreted as interruptive. This information should be mainstream and the preface to every topic explained in text books. I challenge you, if you are not already planning on it, to continue this model for other areas in math. My desire to learn math was sparked not for an affinity to be able to crunch numbers in my head, but rather my fascination with patterns and visualization. Actually, by any standard I'm average at best with mental math, but achieve above average results in mathematics. Calculating is a non-intuitive chore where as visualization exercises tap into, what I believe is, a core skill that all humans have. That being the case, this model has the potential to make math literacy far more accessible.

[@error.418]

I would argue that those who do the best in math competitions use visualization techniques. Visualizing is an important part of math literacy and is unfortunately not widely taught with any quality.

[@arsenalfanatic09]

@@error.418 It feels like the people who are better at math are better at visualization. It can be taught but not many bother to teach it so i think too many people are doomed to thinking they're inherently bad at math

[@error.418]

@@arsenalfanatic09 yeah :(

[@Hexanitrobenzene]

Well, to counter this point, many scientists say that the power of mathematics lies in its ability to help solve problems which are too difficult to visualise and/or where intuitions (including visual) break. That being said, it's always good to have a visual intuition where it's possible.

[@rotemperi-glass4825]

Thanks so much! your videos not only make my view of the world much more interesting and deep, they are also the most fun content I can find on KZbin.

[@kjekelle96]

0:00 intro 1:20 effect of linear transformations on spans 2:59 examples of eigenvectors 4:04 applications 5:15 goal of this video 5:26 how to find eigenvectors and eigenvalues 7:35 geometric meaning of the formula 9:28 revisiting an example 10:46 are there always eigenvectors? 13:03 eigenbases 16:28 puzzle and outtro

[@KaraHinamori]

I feel like part of the reason why your videos work so well is that you give the listener time to pause and think. Even the small pauses after every sentence gives me time to absorb the information, not to mention it's really calming

[@PrakashPant-n8w]

truee... I feel that in his every video.

[@marianseidl662]

I rarely comment on videos but I just have to say this one thing. You deserve so much respect for what you do and how you do it! In all my years of school and university, I never came across anyone who could explain and visualize topics the way you do it. Our world has all these great scientists who discovered unimaginalbe things, but this wouldnt mean anything if we didnt have people like you!

[@HAAH999]

I can't believe that I've spent all these years at school and university without knowing all these things about linear algebra. Specially after this video and knowing the power of eigen basis. Thank you so much for this wonderful series it's actually helping me in my computer vision course. I would be very very grateful if you put another series about Fourier series and Fourier transform

[@NZwaal-tg8ur]

I second this request!

[@grekogecko]

I third it!

[@lated2222]

I'll 2^2 it!

[@danwang4625]

5

[@MsStefHipHop]

I (squareroot of 72)/(squareroot of 2) this message! Could you please post a series on Fourier series and Fourier Transform?

[@car_pal]

this kind of math can only be explained clearly with visual examples and animations it's been more then a week since i started studying eigenvectors and never understood it. Now i'm 3:40 minutes in and i got it lol

some people spent years before this video, no worries, a week is a good time

[@zenchiassassin283]

Well me I don't study it yet but it's interesting

[@niemandniemand2178]

your just an dumbass

[@soundninja99]

@@niemandniemand2178 Piss off. This is hard for a lot of people.

[@squeakybunny2776]

@@niemandniemand2178 said by the person writing 'your' instead of 'you're'

[@meghmalharmanekar1211]

First time in my life I got the insight of what the "diagonalization of matrix" actually means. Heavily indebted to your efforts! Can't express my gratitude.

[@marioalbanese9993]

i've learned more in this 17 minuts than in hours passed at the polytechnic of milan, thank you

[@giuliad223]

FRA 🥺 in bocca al lupo per la sessione

[@javiercomyn6667]

Same but at the polytechnic of Madrid lol

[@hurtihurti]

Same but at the polytechnic of Lausanne haha

[@lucamonegaglia8579]

un fratello

[@Xhemalg]

Vedrai de'

[@alyoco1156]

why am i crying watching these videos. They are so logical that i feel emotional now

[@howardOKC]

I am crying too! I am in tears!!! I love math and love great math learning materials. I just love it!!!!

[@chunlangong2214]

@@howardOKC me too

[@azra8366]

me too, such a good explanation I wish my prof is that good...

[@jerrys_milk]

I'm not crying, but my heart is beating like crazy lol

[@82Muntasir]

Me too dude

[@Buckybarnesfan22]

Came here to revise Eigenvectors and Eigenvalues and ended up watching the entire Linear Algebra series. You're a true legend. Thank you for the clear teaching!

[@guggi_]

Same here

[@ayandas8299]

Same bro

[@PrakashPant-n8w]

same here man

[@SantiagoRodriguez-bx6jy]

you deserve heaven more than anyone

[@gergananikolovagery5058]

oh, man XD yes! you're so right

[@eoe196]

As it is written, There is none righteous, no, not one:

[@vaydada]

Exactly :))))

[@rj-nj3uk]

But he don't want to die.

[@DanielInfrangible]

I'm going to solve quantum computing just so that I can create a real heaven for this majestic animal's brain-soul to be uploaded to after he dies.

[@robertbazzocchi5385]

14:25 *stops video* *plays video two weeks later* I see your point...

[@zairaner1489]

lol

[@DarkFoxV]

I see what you did there... actually no, try to explain it Dx

[@mahmoudrateb7344]

hhahahahhaa, smarter than to try

[@xl000]

I will use your hi res profile picture for something.. Not sure what

[@don9526]

U dont know how to use a calculator?

[@techiesithastobetechies.8531]

I have sunk in more than 7000 hours playing video games throughout the last decade, these videos are more ENTERTAINING than all of those video games. These videos are the most FUN I have had in a FULL DECADE. The amount of "aha!" moments is so satisfying! Feels like I could have invented Linear Algebra all by myself now!

[@charlesz88]

For anyone who is confused about the last exercise: 1. Use NewTransform = inv(EV)*A*EV to get the diag matrix representing transformation A in eigenbasis system. 2. Compute NewTransform = NewTransform^n 3. Use to EV*NewTransform*inv(EV) convert back to the previous system.

[@ompatel8091]

Thank you! It was a bit confusing but your comment made it crystal clear.

[@antonisskarlatos5202]

Thank you too man, it was helpful ! I am just adding some extra details in case anyone needs more help. We have a vector v and we want to apply to it k times the transformation A: A^k * v We know that inv(EV) * A * EV = D, so by mutliplying from left with EV and from right with inv(EV) we take A = EV * D * inv(EV). Now lets try to take A^2 = EV * D * inv(EV) * EV * D * inv(EV) = EV * D^2 * inv(EV). Inductively this gives us that A^k = EV * D^k * inv(EV). So to calculate A^k, we can just in O(n) time to calculate D^k and just apply in the end the two other mutliplications.

[@antonisskarlatos5202]

@@MengLiu-bi9dz At inv(EV) * A * EV = D, we are interested to create the diagonal matrix D. The idea here is that if someone gives us the i-hat = [1 0]^T or the j-hat = [0 1]^T, we would like the transformation D to just stretch them. So let's say someone is giving us the i-hat = [1 0]^T, then through the EV we would convert it to one of the eigenvectors. After we would apply the A and we would get a scalar of our eigenvector, and finally with inv(EV) we would go back, having a scalar of i-hat (remember that the transformations are linear and scalars stay on the same span). So after the three transformations our i-hat, will be converted to something like [k 0]^T, where k a real number. This means that: (inv(EV) * A * EV) * ([1 0]^T) = [k 0]^T so we can easily conclude that the first column of the matrix D is [k 0]. Finally, applying the same thought with j-hat we can prove that D is diagonal indeed. To say the truth, this is the only intuition I do have. My understanding is that we are just trying to get a diagonal matrix. When we manage to get it, the calculations are easy, so with simple algebra we take a close form of A which depends on D and because D is easy to manipulate, A becomes easy as well. I would love to hear other views on it from other people who are more familiar with linear algebra !

[@alphonseraynaud976]

but how do you compute the inverse?

[@forthrightgambitia1032]

I found the chain of logic easier if you convert the basis vectors into the equivalent [1+sqrt(5)/2 1] and [1+sqrt(5)/2 2], then the eigen values pop out.

[@ashutoshpatidar3288]

The Level of Clarity in the words this man spit is absolutely feels insanely Divine!!! Omg is it even possible for someone to be that clearly understandable...he is definitely a miraculous teacher i had ever seen in my life!

[@AntonKimS]

I could not understand eigenvectors and eigenvalues for 14 years. After watching (in utter amazement) all of your videos in just two days, I have finally understood these concepts! So grateful! Thank you!!!!!!

[@Hivlik]

i love when the pi students get mad

[@enormousmaggot]

They always chill back out in the end

[@udaykadam5455]

It bothers me somehow when they show anger instead gratitude

[@EvilMAiq]

@@udaykadam5455 I think it's more frustration than anger.

[@danialdunson]

@@EvilMAiq yeah its more of a table flip react

[@alinapostol2230]

rofl =))

[@baguettegott3409]

This series is so neat. I've watched it a while ago, before learning any linear algebra beyond the absolute basics, and I enjoyed it well enough - although I didn't take that much away from it. Now that I'm actually hearing linear algebra lectures, I regularly come back to particular videos when the topic comes up, just to build up some more familiarity and visual intuition, and I can hardly express how helpful and rewarding that is :)

[@ahmedayman8369]

This...is breathtaking. Mesmerizing to look at these transformations. Dreamy to ponder what those lambdas do and what an eigenvector is. They come to life when I close my eyes now. A very sincere, appreciative and kind Thank You from a struggling student at the University of Hannover.

[@FlareGunDebate]

My experience with math is: watch Khan, watch you, interpret painfully dry book. Thank you, sir.

[@dankazmarek1259]

so real...I feel the same, I am going for Computer Engineering and I know I have to feel the pain of translation and interpretation strike like a thorn in my butt

[@FlareGunDebate]

​@@dankazmarek1259​I've been building physics simulations. Most pages are self advertising in disguise as educational resource. That or hobbyists publishing their inner thoughts and muddled process. I hope to write some clear articles on the subject one day.

[@FlareGunDebate]

@Dr Deuteron I've done the math. Khan Academy is good for working though that. This channel is good for the intuition and thought experiments.

[@sukhmandersingh4306]

Mit courseware is also very good.

[@animeshsinghal3405]

At 3:50, I paused the video and celebrated my excitement for 5 minutes. THIS MAKES SOOOO MUCH SENSE!! The build-up was worth it! Thank you :'")

[@andrewmerafuentes6683]

Why do most of us pause around this time?

[@gerardsagliocca6292]

Listening to this video is the first time I actually understood what an eigen vector and eigen value really means because you gave the visual representation of that an igen vector, eigenvalue is doing on a x y plane. No textbook that I ever bought or borrowed at a library ever showed your graphical meaning. The authors went on and on about how to find them but never gave the student to he graphical dynamics involved to get that quick realization. Even MIT professor Strom I believe never showed any visual presentation, so nobody really understood what was going on in linear algebra and so linear remains a scary topic in mathematics for many students. So I am glad I happen to come across this inspiring video that wiped away all the fear and anxiety over a required course in most tech curriculums. How you figured out how to fix this awful situation is truly an amazing thing. You seem to have a gift of clarifying some reALLY NASTY situations in mathematics. Kudos to you. And while I am at it, you also clarified quickly confusion in another topic in mathematics that electrical curriculums discuss but never really clarify what it really means and that is ...Convolution ! Today, in 2023, students are fortunate to have great videos on KZbin so they can. Get away from technical books that never clearly explain anything, except having many problems at the end of a chapter which many students can't do because textbooks are a 2 dimensional format and most times one needs a 3 dimensional tool to explain the graphical interpretation so students can quickly understand the topic being discussed. So I am glad textbooks are being replaced by more better tools to convey the meaning to a student trying to learn the math and the concepts being introduced by a teacher. ,

[@dn9255]

This video gave me so many "AHA!" moments and cements all the information you've taught in former videos of the series. Thank you so much!

[@noahmarshall4520]

This series is literally worth more than the 400 I've paid to take linear algebra in uni.

[@diegomastro5681]

Then donate

[@soundninja99]

Currently taking linear algebra in uni (lin. alg. for engineer students, no the more proof oriented one for math students). This is a great supplement to lectures to truly understand the material, but it doesn't replace it. Super fucking grateful for this though. Grant makes maths more fun

[@davibergamin5943]

Currently taking linear algebra in uni, even my teacher recommended this serie

[@PJMM]

@@soundninja99 This, thank you! I read it so often that "this is so much more worth than university". But IMHO and personal experience, I believe I wouldn't grasp the entire concept in its depth just by KZbin videos, most of all not as complete and structured as taught in university (at least over here & it might be worth mentioning that it's free) and most of all I'd lack the learning environment created at this place. But you are also spot on about as supplement material, this is really invaluable. And I do still get where many people are coming from - 3b1b really does take the time to explain the fundamentals in a way it's often rushed past at university, leaving you behind with the feeling of "not really getting it"! So all in all, you put that into much better words than I could anyway. Much appreciated, really.

[@jasonlewis5125]

I feel like I’m gonna cry. The detailed visuals and pauses while explaining things show that you care about us understanding. I’ve never felt someone care so much about my understanding to pause like this. I know it’s just a KZbin video but thank you!

[@patrickwienhoft7987]

This was probably the biggest enlightening I experienced ever...

[@drdcs15]

I look forward to these every day, hoping one will come out. I've tried so hard on my own to understand all of this. It's like I have a ton of almost finished puzzles floating around in my head and every video I watch a piece clinks into place and the one of the pictures is revealed. Absolutely incredible. Thank you

[@Spartacus005]

The amount of times I've yelled "Oh my God I get it" so far is astounding and I'm 3:33 in. I can't wait to find out what I'm yelling it next for! I just learned I needed to know how to compute eigenvectors and eigenvalues for my seismology class, and seeing this video has lightened my day considerably! Thank you thank you thank you so much for creating these wonderful, educational videos!

[@andresmassigoge8600]

From the thousands of Eigenvideos on youtube, this is truly an Essential one.

[@guigagude]

Andres Massigoge It's the eingenbasis, for sure

[@motaaaa]

Astounding, I'm going to study this subject next semester and it's wonderful how I can already grasp it's intuition quite well, you sir deserve some 1 billion subscribers

[@jnayehsirine6222]

8 years ahead , and this still the best serie for Linear Algebra basics ! thanks sirr

[@Oreoezi]

You just turned 1 hour of university in 17 minutes of things I actually understand. Thank you so much.

[@MsAlarman]

Thanks!

[@lucasnebelung5430]

I never really thought of Maths of something fun, but your videos make it so easy and most importantly fun to understand all the concepts and how they are actually closely related to each other. I'm so thankful for your videos and really enjoyed watching all of this and your other series on Analysis etc. You're by far the best math teacher and in my humble opinion a million times better than anyone else on YT. Keep up the great work. Thank you so much!

[@germanbuenolozano3941]

Good eavning, I am german, an engineer on formation, I feel the obligation of thanking you for this video, I am going to pass my test thanks to you

[@altolegato]

I have to point out a nice trick about the eigen stuff. If during exam, you obtained all eigen values for a matrix in previous questions, and the next one requires the DET of the same Matrix, Please note that The DET of that Matrix=Product of all eigen values. It saves your time during exam.

[@zairaner1489]

There actually is another way which just needs you to have calculated det(A-lamda*Id), the determinant will be the constant part of the resulting polynomial

[@sanchitapihu]

I had never ever come across such a beautiful explanation of eigenvalues and eigenvectors. This is by far THE BEST explanation of the concept. The entire series is mind-blowing. Never saw matrices from such a perspective. Hats off!!!

[@gumball135]

SPOILERS. Here's what I've discovered about the puzzle at the end. Observe that squaring A gives successive elements of the Fibonacci sequence F_n, so A^n = [[F_n-1, F_n], [F_n, F_n+1]]. An efficient way to compute A^n will also give an efficient way to compute F_n.Take the eigenbasis E = [[2, 2], [1 + sqrt(5), 1 - sqrt(5)]]. Now the matrix B = Einv * A * E gives a diagonal matrix, as you see in the video. It's easy to compute powers of this matrix, B^n, by squaring the elements. Taking the nth power of matrices of this form is actually equivalent to squaring the matrix in the middle and then multiplying by the matrices on the left and right, since B^n = (Einv * A * E)^n = (Einv * A * E) * (Einv * A * E) * ... * (Einv * A * E) = Einv * A^n * E. To understand the last step, note that the Es and Einvs cancel each other out when you rearrange the brackets. Finally, we can multiply B^n by E and Einv, and out pops A^n: E * B^n * Einv = E * Einv * A^n * E * Einv = A^n. Which gives us the nth Fibonacci number. (Edit: corrected typo in A^n).

[@srinmex]

Thanks for your explanation! I think you have a typo A^n = [[F_n-1, F_n], [F_n-1, F_n+1]] should be A^n = [[F_n-1, F_n], [F_n, F_n+1]]

[@liles337]

This is an excellent explanation. Thank you for sharing.

[@FlyingSavannahs]

Yes, a Fibonacci sequence emerges from the multiplication. I got A^n = [ f_n-1 f_n ] f_n f_n+1 where f_n is the nth Fibonacci number. You didn't specify the values of your matrix B. I got values involving the golden ratio, i. e., B = [ gr 0 ] 0 (1-gr) Grant states that transforming B^n back to get an interesting function, but I don't follow your process here. Any thoughts?

[@gumball135]

@@FlyingSavannahs I haven't watched this video since I wrote the comment, but I think he just means removing the E and Einv at the end to get A^n and the nth Fibonacci number. This requires 2 matrix multiplications, which are equivalent to linear transformations, hence why he refers to it as a transformation. The whole process is a function of n, g(n) = E B^n Einv = A^n. Does that clarify anything? 😄 Like I said, I haven't rewatched the video, so I might be completely missing your question.

[@Iceiceice2023]

@@FlyingSavannahs this is confusing to me too, it seems the calculation become harder to calculate the orthorganal Matrix, since you have to calculate the power of 1+sqt5 to n. Did you figure this out?

[@LMau-t9r]

I just had this determinant class, you explained perfect what eigenvektor and value is as well as why is det(A-λI) even used, thank you for saving me hours of my life

[@gregoryneedsaname]

Wow. It only took you 3 minutes to explain something that I couldn't understand for the past 23 years. Bravo!

[@maartenkeus8627]

In literally the first 10 seconds I have already gained a better understanding then uni could have evert taught me, you're an actual wizard and these visualizations are revolutionary Thank you

[@huanyichen8492]

@16:30 My general idea is First you perform a change of basis by doing D = E_inv * A * E, where D has to be a diagonal matrix of eigenvalues [[lambda_1, 0], [0, lambda_2]]. Then performing the A^n under the new basis will be the same as stretching the eigenvalues by n times, which gives you M = D^n = [[lambda_1^n, 0], [0, lambda_2^n]]. Last you need to change the basis back, which can be done by doing M’ = E * M * E_inv. Then the M’ will be the answer you are looking for.

[@jeffery_tang]

wow thanks i got it! A = [ 0 1 1 1 ] E = [ 2 2 1 + sqrt(5) 1 - sqrt(5) ] D = [ 1.61803 0 0 -0.61803 ] B (D^10) (B^-1) = [ 34 55 55 88 ] approximately, lines up with fibonacci as well

[@killianpinier5074]

Thank you so much for your incredibly rich content. Unlike most professors, you start by explaining the practical interpretation of a concept before translating it into theory. This approach is refreshing because many people are satisfied with just understanding the theory, but they often miss out on its physical meaning. This gap is why many struggle with physics: they learn the theory but don’t know how to apply it to the real world. But solving a problem requires working backward: you interpret the real world and apply it to the theory.

[@henriquegomes100]

I love the dramatic phrases on the begging. It's nice to see someone who loves mathematics so deeply.

[@Zeus.2459]

11:30 hit like a ton of bricks I paused the second I saw "i", and thought back to his video about euler's identity maths is goddamn beautiful

[@lord_napoli]

xd

[@OktatOnline]

Around 7:50 as the 2 dimensional space was "spinning" and being squished into a lower dimension, I couldn't help but think that this is as if the "sheet of paper" was being rotated around that same 3-dimensional axis as the one you mention around 4:30. World class explanations, helps me so much! I saw that you have a whole playlist of these, subscribed immediately.

[@wiaschtlsiara9516]

Astonishing animations, perfect explanations, high quality audio. Nothing my university has. Thank you very much.

[@tokkia1384]

“Squishification” 😂 ❤️ made my day

[@hemrajpandeya81]

me to

[@Amb3rjack]

Yes! That word could well be the key to life, the universe and everything. For me it really is a genius piece of fully understandable made up language that exemplifies the genius of this gentleman's concise but very easy going and extremely watchable teaching methods. Just magnificent!

[@winterturtle1596]

Sometimes in my senior level undergraduate numerical methods class I get confused, and I keep coming back to this video. It's such a good way of understanding these concepts. To me, the most useful parts of this are definitely the showing mathematically why the formula Av=lambda*v comes from and how it relates to the method of finding eigenvalues, as well as the change of basis formula in relationship to achieving an eigenbasis. Interestingly, as we learned in this class, you can solve for the eigenvectors by looking for a matrix such that when used as a change of basis it results in a diagonal matrix for any matrix A. Thanks to your video, statements like this aren't astounding, or something I would need to memorize, but rather something that is obvious, and intuitive. Thank you again for these highly educational videos, you are doing a great service to the world.

[@johnhumbug5199]

lol says: i won't explain the computation in detail! But explains it by making the computation as intuitiv as possible. Thanks for this series...

[@SmileyMPV]

That puzzle at the end is basically a very complicated way to get the fibonacci formula... AND I LOVE IT

[@swatantradhara4153]

Spent an hour reading the book: Didn't understand a thing. Watched this video till 6:54: Understood everything. I really wish I was taught this way back in school! Love your materials!

[@isaiasgarcia8687]

11:34 The mathematician that first came up with "imaginary" numbers refered to them as lateral numbers, as saying a number that doesnt belong to the Real numbers plane. I'm pretty sure that in this case, the eigen vector is the one that pops out of the screen, the axis of rotation, just as in the 3d model.

[@martinepstein9826]

The eigenvectors are [i, 1] and [-i, 1]

[@pranavkondapalli9306]

i'm pretty sure euler was that mathematician

[@imonbanerjee2997]

You are right, but also "not quite". What you described is R^3 or the real space. Let us take the example of complex numbers vs. the real plane. As you described, there is a direct relation between them. (y-coordinate maps to the imaginary number). However, there are many distinctions. (I am really trying hard not to just throw "diffeomorphic functions on the complex plane are analytical whereas functions on R^2 don't have the same property" because the terms might not make sense.) But take a look at this example. (1,0) and (0,1) are two elements of R^2. Now, take the corresponding elements in C 1 for (1,0) and i for (0,1). Note that, if you take the products, in the first case, you get (0,0). However, in the second case. you get i, whereas you should have gotten 0. In fact, for any two complex numbers z_1,z_2; z_1z_2=0 only when one of them is 0. Such restrictions need not hold for multiplications in R^2. Take another example, you can divide 1 and i and get -i. However, you cannot divide (1,0) and (0,1) to get (0,-1). In other words, you can multiply i with -i to get 1. However, you cannot multiply anything scalar to (1,0) to get (0,1). These are some examples where R^2 and C differ

[@NitinNataraj-gf3vx]

This is so succinct, and just simply brilliant. It is helping me get through my Master's degree in AI and I can now see everything intuitively. Thanks a lot for these! I have asked all my friends to subscribe!

[@triton62674]

Maths degree in AI? Which university?

[@Prometheus7272]

triton62674 He said masters

[@stanpreschlack6196]

i cannot believe how you explain these concepts so well never in a million years did i think i could understand linear algebra but watching your videos all of the concepts just 'click' and it makes it so easy to learn more about the topic because you offer such an effective framework of understanding.

[@bengski68]

At 4:40, it seems like you're brushing the possibility of an eigenvalue of (-1) under the rug. Presumably, we need some argument to the effect of "rotations are orientation preserving, and therefore have positive determinant. All non-stretching/squishing transformations in 3D (or odd dimensional) space have an eigenvalue of 1".

[@3blue1brown]

Good observation, I failed to bring up the 180 degrees edge case.

[@bengski68]

Also the case of reflections, which flip but don't stretch or squish.

[@gorgolyt]

Reflections are not a case of rotations...

[@bengski68]

Ryan Denziloe But the line of reasoning was _"because a rotation doesn't stretch/squish, it must have an eigenvalue of 1"_. My point is that the same line of reasoning could be used to lead to the _false_ conclusion that any reflection has an eigenvalue of 1.

[@gorgolyt]

bengski68 Okay, I understand your point now. The fact it's not -1 is indeed implicit in the argument, although if the viewer has understood what eigenvalues are, it should be clear what's meant.

[@saitaro]

I like the indignation of the little pi's animation :P

[@ntcpizza7452]

This series and your channel have taught me to love math for its sheer power. Thank you for bringing this into my life

[@ophello]

The imaginary eigenvalues blew my mind. That's where euler's identity comes in!

[@Alzeranox]

Funny enough, there is an vector which is going completely unchanged in that example. But it extends into three dimensions. If you go back and watch the example, imagine a line coming straight out of the origin towards yourself. That imaginary line is the eigan vector.

[@AuroraNora3]

Timothy :0

[@Raikaska]

Timothy but arent we sitting in 2 dimentions? We could also argue that E5, E58 remain uncanged, but they arent in the original space? I guess its related to cross product

[@ゾカリクゾ]

we are in 2D going back between dimensions is not so simple. what you are saying would make our vectors something like and , etc. (with z = 0). That would be 3D, but the vectors seen in screen were all represented by 2-tuples, therefore they are 2D.

[@BettyCastella]

@@Alzeranox which corresponds to the imaginary axis!!!

[@sheepphic]

The next one is the last one? Nooo! I was enjoying this series so much!

[@lifestyleam-k3d]

Thank you so much for creating these videos! As a university student, I often find textbook materials not enough for visualizing linear algebra concepts. Your visualizations bring these abstract ideas to life, making them much easier to understand. Last week, even my professor put your video during our office hours session and advised to watch all of them during summer. Your efforts are truly appreciated-keep up the great work!

[@TheFarmanimalfriend]

I had 'weak' background in math when I first encountered them in quantum theory (chemistry). They almost blew me out of the water! I wish I had had access to this kind of video back in my undergraduate days. Mathematics is really cool.

[@gustavopazrequejo6691]

OMG! I'm in 4th year of the degree physics and at the min 3:38 i started to cry

[@anahitaabdollahi4584]

I'm a second year physics major and I'm crying too :)

[@matron9936]

Im in seventh grade and I’m ain’t crying :)

[@matron9936]

r/iamverysmart

[@MCMasters4ever]

@Lea I'm only half american and I only shed a few tears!

[@ozzyfromspace]

I'm a dropout, so I smiled

[@haris7247]

A very neat explanation by some guy on mathstack exchange: Eigenvectors make understanding linear transformations easy. They are the "axes" (directions) along which a linear transformation acts simply by "stretching/compressing" and/or "flipping"; eigenvalues give you the factors by which this compression occurs. The more directions you have along which you understand the behavior of a linear transformation, the easier it is to understand the linear transformation; so you want to have as many linearly independent eigenvectors as possible associated to a single linear transformation. Consider a matrix A, for an example one representing a physical transformation (e.g rotation). When this matrix is used to transform a given vector x the result is y=Ax. Now an interesting question is Are there any vectors x which do not change their direction under this transformation, but allow the vector magnitude to vary by scalar λ? Such a question is of the form Ax=λx So, such special x are called eigenvector(s) and the change in magnitude depends on the eigenvalue λ.

[@zairaner1489]

What? The shock! I didn't expect that so quickly! I'm not prepared!!

[@alvarol.martinez5230]

I just wish he is saving material for an Essence of linear algebra II

[@marinarapp500]

I have many mindblowing moments watching this series. Makes me like maths much more! Thank you!

[@henryktocoaching]

Wow, I got my undergrad in mathematics 25 years ago - never could get the visuals behind the mechanics of eigenvalues and eigenvectors. I veered away from math after I graduated and went into finance/investments, but now with the proliferation of AI, find myself digging through the mathematics again and this was super helpful. Thank you for doing us a service!

[@clavisjanes3927]

All of your videos are so thorough; truly amazing!

[@eduardogomes4865]

Could you elaborate on 11:34 in another video? I would like to see a connection between vectors and complex number as they seem to have very similar applications

[@zairaner1489]

There are multiple ways you can look at the connection between 2x2 matrices and complex numbers and it always has to do with 2x2 matrices of the form a -b b a One way is to literally define the complex numbers as the set of all 2x2 matrices of this form. Then the normal addition of complex numbers correspond to matrix (componentwise) addition, complex multiplication to matrix multiplication, the real part of the matrix is a, the imaginary part is b, one over a complex number corresponds to the matrix inverse, the determinant of the matrix is the square of the absolute value, transposing the matrix is conjugating the complex number. Also notice that the imaginary number i corresponds to the matrix 0 -1 1 0 the matrix shown in the video to have eigenvalue +-i. The other way to look at these matrices is to look at them as the only linear transformation which do not only have the scaling property regarding to real numbers, but even for complex numbers, so they satisfy A(z*w)=z*A(w) for all complex numbers z and w (here regrading the ocmplex numbers as a 2D vectorspace over the reals). This happen to be exactly the functions f(z)=a*z for a constant complex number a, so the linear transformations which correspond to multiplication by a complex number a. The matrix in the video corresponds to multiplication by i-which has ofc eigenvalue i. If thats too much: Just remember that multiplying by a complex number always corresponds to a combination of a rotation and scaling in the complex plane. Multiplying by i rotates everything by 90 degrees and thus the linear trsnformation for a 90 degree rotation and i naturally correspond to each other

[@eduardogomes4865]

Raphael Schmidpeter Thanks very much, I was really looking towards the matrix representation of a complex number, as until now I had only seen the vector one. Just one more thing: If a matrix a -b b a Corresponds to a complex number z, and the linear transformation that occours when multiplying by it, does that means its eigenvalue will be that number, z?

[@zairaner1489]

Eduardo Gomes Exactly (and -z as the other eigenvector)

[@eduardogomes4865]

Raphael Schmidpeter ty, that helped me a lot

[@ir2001]

Feels good to hear such discussions

[@utkarshshrivastava4382]

Beautiful Video man, I watched the whole series for this video, and it was one of the most clarifying experience I have had for linear algebra. I have been asking everyone, seniors, friends, professors, google, everyone, what do eigenvectors and eigenvalues mean, and this is the first time after 6 years, literally the first time that I have finally understood the depth and meaning of these things. It was a mind = blown moment when I watched it after 14 previous lectures in this series. Thank you for making this whole series, and kudos on your work. Your style of teaching, ease of explaining, clarity of concepts, beauty of animation and the simplicity of music and videos is extremely commendable. I genuinely think that this whole series should be taught as a part of linear algebra courses across all colleges around the world. The finest level of clarification students will get will help them in every aspect of their studies, research, way of thinking, everything. Again, thank you for making this series, absolutely loved it, and I at least will be referring to your channel for any clarification/assistance in topics related to mathematics. Good Job and Good Luck!!!

[@panayiotismichaelides3148]

Super intuitive and well explained, amazing video!

[@wadoichimonji8522]

Back in school, I was made to memorize different types of matrices, I always wondered why they were all so necessary, especially this "Diagonal Matrix", I wondered what's the big deal about the diagonal of a matrix anyways and now about a decade later I finally have my answer. Thank you Grant.

[@SurabhitGupta]

Thanks! I was revising maths for my PhD in Mechanical Engineering and I wish I watched this video earlier.

[@neerajaggarwal4689]

Just imagine how much difficult it is to teach topics like these on a board. You can blame your teachers but just imagine.

[@ellyvatedaf]

Well, thats why its important for students to sit down with their own thoughts and time to visualise whats happening by themselves

[@MilosMilosavljevic1]

Why would they have to use a board though? Any decent math professor should be able to use something like Mathematica or Geogebra to produce a decent animation to suplement a lecture. So the problem is not in the difficulty of how, it is almost certainly in the lack of why. There simply is no incentive to be a great lecturer when in most universities you are only judged by your research. You do your research well enough and you can be the worst lecturer ever, your job is perfectly safe. So, most professors will look at making an effort to produce great learning materials as a waste of time, if they focus too much on lecturing and lag behind in research, they might easily lose their job to someone that outproduces them in research.

[@user-jh3kz7dp2z]

@Arjun lalwani GeoGebra, it's the name of a software, you can't just casually rescript it

[@umadbroyo2388]

@@MilosMilosavljevic1 Spoken like a true student from a first world country!

[@PaulSmith-pr7pv]

@@MilosMilosavljevic1 academia is stupid

[@guycollishaw1957]

The moment I hear the word Eigen... my brain just decides to work at 10% its usual capacity.

[@enormousmaggot]

Once you finish this video, go check the German too -- eigen literally just means 'one's own'

[@CuteLittleHen]

Think about Eigenvalues as being a self value. :) For example, if the vector spaces has a self value of 0, this means the kernel is not made of only the 0 vector.

[@khatharrmalkavian3306]

I feel that way about sigma notation.

[@tanmaydeshpande]

The moment I heard the word Eigen, my mind went straight to the Endgame scene where Tony discovers Time Travel

[@alexandertownsend3291]

@@khatharrmalkavian3306 sigma like for standard deviations or do you mean the one for sums? I can explain whichever one you want me to.

[@jessicathelen7810]

I can't wait to watch the rest of these! I am currently in LA again as a refresh and my instructor did not teach it well the first time and unfortunately am in the same boat again! I literally got up at 3 1/2 minutes and just paced around b/c it blew my mind w/ understanding - FINALLY! Halfway through, I paused and shared it with my college class who is also struggling! This one video helped me so much already seriously - thank you!!!!!!!!!!

[@neonlight1203]

Thank you for making me ask questions like these: "If the eigen vectors of a matrix are orthogonal, then is the product of the eigen values equals the determinant of that matrix?" More appreciation if someone explains this.

[@harshkn]

I was waiting for Eigen vectors video since you started this series. Thanks! Appreciated :)

[@Hevever]

Did an undergraduate degree in mathematics and yet this is the first time I have thought about these concepts in this intuitive way..! could do the sums but never understood what was going on behind the scenes. wish I had had these videos during my degree but glad to see them now!! thanks so much

[@gregoriofreidin4683]

i must be watched this video like 10 times during my career, always love how he explains

[@maurolarrat]

Could you please increment this serie with Singular Value Decomposition, witch make use o eigenvectors and eigenvalues? Thanks.

[@amit4422]

I would request the same as well. I hope you extend this playlist with 1 last video regarding SVD.

[@area51xi]

Incrementing this request.

[@luisleal4169]

same request here

[@rusho906]

Requests++

[@austinyhc]

++ please

[@mark7166]

Linear algebra was always one of my favorite subjects back in my engineering education days. I'm relearning it as part of an effort to train myself in machine learning, and this series has reminded me of exactly why. It's an astoundingly beautiful topic.

[@evangelosstromatias4879]

This guy is the Morgan Freeman of maths. Thank you!

[@FlyingSavannahs]

No. Morgan Freeman is the Grant of acting!