Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra
Вставка
- Опубліковано 14 вер 2016
- A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
Help fund future projects: / 3blue1brown
An equally valuable form of support is to simply share some of the videos.
Home page: www.3blue1brown.com
Full series: 3b1b.co/eola
Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.
3b1b.co/support
A solution to the puzzle at the end:
www.dropbox.com/s/86yddvprfua...
Typo: At 12:27, "more that a line full" should be "more than a line full".
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate.
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown
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!
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.
Yes your generation is very spoiled.
True! When i went to school.....NEVER explained like this!
When I took linear algebra, I was not able to comprehend exactly what the eigen value and eigen vector represented. I just memorized the formulas. We truly live in an exceptional time. (I didn't take advantage of youtube as a resource for learning until recently)
@@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 :)
"I wont teach you how to compute them" - Proceeds to teach us how to compute them better than any textbook or professor ever could
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.
@@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"
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
@@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.
@@gracialonignasiver6302 same here :)
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.
Thanks so much!
@@3blue1brown changing lives dude!
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
@@wesm6747 Where do you use Eigenvectors and Eigenvalues in Computer Engineering if you don't mind me asking?
@@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.
you deserve the nobel prize in maths for making math accessible like this to millions of students
Since that there is no math nobel prize, a Fields Medal should do the work. And yes, Mr. Grant deserves it!
Totally agree
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]
there isn't a nobel prize in math
I wish I had this in college. I struggled with this subject so much
Thanks! Hopefully, current college students find it helpful. By the way, just watched your transistor video and loved it!
They definitely will and thank you!
Hey, glad to also see you here, love your videos!!!!
CGP grey should also visit. Actually this 3b1b voice sounds like greys ..
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 UA-cam/academia (pretty sure they're considered equivalent).
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 :)
Definitely
fax
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".
Thanks!
Same in Dutch
In brazilian portuguese, we call them "autovalores e autovetores", which would sth like own-values and own-vectors, respectively...
😊
In french is "vecteur propre"
Of course english is the only language that leaves it in german.
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.
What kind of monster would downvote this masterpiece? This may very well be one of the best series ever made.
Robert
What are you talking about?
The votes are overwhelmingly positive.
Ssshh!
It's people from Australia and New Zealand.
@@alexander-jl6cs Oh yeah, their votes scale with an eigenvalue of -1
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)
Probably those who say “i hate maths”... ;)
i love when the pi students get mad
They always chill back out in the end
It bothers me somehow when they show anger instead gratitude
@@udaykadam5455 I think it's more frustration than anger.
@@EvilMAiq yeah its more of a table flip react
rofl =))
It's so lucky that this series is already complete when I'm studying linear algebra
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]
@@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.
@@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".
@@floatoss thnx , now I got it
yeah
i've learned more in this 17 minuts than in hours passed at the polytechnic of milan, thank you
FRA 🥺 in bocca al lupo per la sessione
Same but at the polytechnic of Madrid lol
Same but at the polytechnic of Lausanne haha
un fratello
Vedrai de'
I cant name one video producer who has such an enormous positive feedback and with viewers who are so fascinated by the content!
You commenters are the freaking best. Usually, UA-cam comments can be such a dark hole, but every video I've been uplifted and pumped to make more.
+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!
why am i crying watching these videos. They are so logical that i feel emotional now
I am crying too! I am in tears!!! I love math and love great math learning materials. I just love it!!!!
@@howardOKC me too
me too, such a good explanation I wish my prof is that good...
I'm not crying, but my heart is beating like crazy lol
Me too dude
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
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!
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.
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.
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
15 min? are you watching math on 1.25 speed?
sometimes i do in 2X depending on motivation, attention, professor age :)
dumbass
you deserve heaven more than anyone
oh, man XD yes!
you're so right
As it is written, There is none righteous, no, not one:
Exactly :))))
But he don't want to die.
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.
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!
Same here
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.
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
I second this request!
I third it!
I'll 2^2 it!
5
I (squareroot of 72)/(squareroot of 2) this message! Could you please post a series on Fourier series and Fourier Transform?
14:25
*stops video*
*plays video two weeks later*
I see your point...
lol
I see what you did there... actually no, try to explain it Dx
hhahahahhaa, smarter than to try
I will use your hi res profile picture for something.. Not sure what
U dont know how to use a calculator?
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!
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
“Squishification” 😂 ❤️ made my day
me to
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!
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!
You just turned 1 hour of university in 17 minutes of things I actually understand. Thank you so much.
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!
This series is literally worth more than the 400 I've paid to take linear algebra in uni.
Then donate
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
Currently taking linear algebra in uni, even my teacher recommended this serie
@@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 UA-cam 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.
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
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!!!!!!
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.
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.
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.
@@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
@@arsenalfanatic0971 yeah :(
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.
My experience with math is: watch Khan, watch you, interpret painfully dry book. Thank you, sir.
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
@@dankazmarek1259I'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.
@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.
Mit courseware is also very good.
No word can do justice in praise of your great knowledge neither to the efforts you put in to make these animated videos. You are just incredible. The world is in dire need of teachers like you.
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 UA-cam video but thank you!
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
Well me I don't study it yet but it's interesting
your just an dumbass
@@niemandniemand2178 Piss off. This is hard for a lot of people.
@@niemandniemand2178 said by the person writing 'your' instead of 'you're'
This was probably the biggest enlightening I experienced ever...
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 :'")
Why do most of us pause around this time?
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!
From the thousands of Eigenvideos on youtube, this is truly an Essential one.
Andres Massigoge It's the eingenbasis, for sure
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 :)
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
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.
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
xd
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.
Thank you! It was a bit confusing but your comment made it crystal clear.
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.
@@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 !
but how do you compute the inverse?
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.
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 UA-cam 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.
,
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!!!
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!
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
The content is so good that it needs to be seen more than once to understand the deeper meaning of the concepts.
This series needs to be binged several times at least for me :)
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.
I like the indignation of the little pi's animation :P
The imaginary eigenvalues blew my mind. That's where euler's identity comes in!
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.
Timothy
:0
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
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.
@@Alzeranox which corresponds to the imaginary axis!!!
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!!!!!!!!!!
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
Just imagine how much difficult it is to teach topics like these on a board. You can blame your teachers but just imagine.
Well, thats why its important for students to sit down with their own thoughts and time to visualise whats happening by themselves
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.
@Arjun lalwani GeoGebra, it's the name of a software, you can't just casually rescript it
@@MilosMilosavljevic1 Spoken like a true student from a first world country!
@@MilosMilosavljevic1 academia is stupid
The moment I hear the word Eigen... my brain just decides to work at 10% its usual capacity.
Once you finish this video, go check the German too -- eigen literally just means 'one's own'
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.
I feel that way about sigma notation.
The moment I heard the word Eigen, my mind went straight to the Endgame scene where Tony discovers Time Travel
@@khatharrmalkavian3306 sigma like for standard deviations or do you mean the one for sums? I can explain whichever one you want me to.
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.
This series should be mandatory for intro to linear algebra courses. Grant is an incredible teacher and science communicator.
Don't have words to thank the awesome work you are doing... You are lifesavers for a new generation of mathematicians, scientists, engineers, and in general, for people like me, trying to learn something everyday. THIS, is the bright side of UA-cam. Masterpiece!
Astonishing animations, perfect explanations, high quality audio. Nothing my university has.
Thank you very much.
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 UA-cam.
This is probably one of my favourite 3B1B -videos. All the physics using eigenvalues equations feels so much more intuitive now!
This guy is the Morgan Freeman of maths. Thank you!
No. Morgan Freeman is the Grant of acting!
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
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
Super intuitive and well explained, amazing video!
Bravo! The best animations I've ever seen. It's impressive how you explained eigenvectors in less than 20 min. Thank you so much for helping me understand
I have many mindblowing moments watching this series. Makes me like maths much more! Thank you!
i have watched this videos over 30 times in past 6 years and yet everytime i find something new, and always come from trying to understand eigenvalues from adifferent point of view, maths is amazing
this was amazing. saved me a lot of time of struggling trying to visualize everything. may whatever you believe in bless you
All of your videos are so thorough; truly amazing!
That puzzle at the end is basically a very complicated way to get the fibonacci formula...
AND I LOVE IT
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
Math can always brings to mind the wonders of making a concept so understable , so simple ... And make many solutions reachable . Thank you very much, Grant.
I love when the pi-creatures raise their arms and frown like that, it's hilarious
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.
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
I've been doing eigenvalues/vectors in differential equations and linear algebra for a while now, and i always knew *how* to do them, but not *why* they did what they did. This totally changed my perspective on LinAlg, and it's so much more straightforward to do it now! Thank you!!!
Its incredibly impressive how Grant is able to make the flow of information so quickly efficiently and correct.
OMG! I'm in 4th year of the degree physics and at the min 3:38 i started to cry
I'm a second year physics major and I'm crying too :)
Im in seventh grade and I’m ain’t crying :)
r/iamverysmart
@Lea I'm only half american and I only shed a few tears!
I'm a dropout, so I smiled
I love the dramatic phrases on the begging. It's nice to see someone who loves mathematics so deeply.
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!!!
The work you are doing is amazing. I’m so lucky to have found this channel while studing aerospace engineering. Keep it up!
What? The shock! I didn't expect that so quickly! I'm not prepared!!
I just wish he is saving material for an Essence of linear algebra II
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...
Man I have been following this page for years even before even thinking I’d go into engineering in my late 20’s and I always liked these videos at a distance not understanding any of the math but appreciating how satisfyingly these things worked. Now.... I need these videos to understand the horror of the difficulty of the math I know am actually doing.
Would like to thank you for making the subject so accessible to even high school students like myself. I solved the question asked and it was a real treat! Binet's formula appears out of nowhere...just amazing! This series has helped me so much! Thank you once again.
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).
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]]
This is an excellent explanation. Thank you for sharing.
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?
@@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.
@@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?
I almost crying... It's just amazing to feel all the blury data in my brain suddently become cristal clear... I will never thank you enough for making this content for free... I'll tip something to you, i owe you. Thank you, and god bless 3b1b
I am mindblown this has been more informative than a whole semester on linear algebra. THANK YOU
Can we take a moment to appreciate the wonderfully timed pauses in the video so we the audience can digest the information!
My brain is crying tears of joy
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.
what a fantastic visual representation! This entire video series has been a life changer!
using this right now for my linear algebra final. These videos are a HUGE help understanding these topics conceptually. We learned how to calculate these things mechanically, but the visualization and ideas were what i needed to understand better and this is really really great at that. Thankyou.
i must be watched this video like 10 times during my career, always love how he explains
I was waiting for Eigen vectors video since you started this series. Thanks! Appreciated :)
Not sure if you still comments on old videos but my professor for a fourth year CS class assigned this video for us to watch because she said most students struggle to truly understand eigenvalues and eigenvectors. She was right, this is so helpful!
I can feel strange words I've heard over YEARS melting into meaningful concepts in my mind just like that, thanks!