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!

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

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.

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

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

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 :)

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

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".

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

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

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.

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 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.

you deserve heaven more than anyone

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.

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!

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

i love when the pi students get mad

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!

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

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 :'")

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.

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

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

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.

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

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

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 KZfaq video but thank you!

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 KZfaq 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. ,

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!

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 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!

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.

“Squishification” 😂 ❤️ made my day

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

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!

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

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

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.

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

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 KZfaq.

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 :)

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

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.

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!!!!!!!!!!

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!

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

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.

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 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

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

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!!!

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

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

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

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

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

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

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

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

This is probably one of my favourite 3B1B -videos. All the physics using eigenvalues equations feels so much more intuitive now!

I'd probably ace all my math classes if the lecturers actually explained what the heck we're doing instead of writing formulas first thing they're in the class.

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

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

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!

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 KZfaq. Masterpiece!

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

Super intuitive and well explained, amazing video!

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!!!

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.

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

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 :)

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!!!

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

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

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.

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.

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

All of your videos are so thorough; truly amazing!

You explained this whole series so well and thoughtfully that you can go months without doing anything related to linear algebra and still have a clear, lifelong grasp on the most important aspects. It honestly beautiful what you have done. P.S. Your videos helped me tremendously to get a Masters in A.I. and deep learning. Thank you so much!!

This series should be mandatory for intro to linear algebra courses. Grant is an incredible teacher and science communicator.

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

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...

Never really had use for eigenthings, so I kinda glossed over this video the first time I came across it while going through the linear algebra playlist - but really fantastic to come back now that I *do* need to work with eigenvectors/values, and have this outstanding explanation. Amazing work as always! I don't think I'd ever have enjoyed (or been able to even do) math as much as I do without 3b1b videos

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

Thank you, thank you, thank you! I've been needing this explanation for years. This was incredibly helpful.

My brain is crying tears of joy

Amazing! After going through several text books and articles, I finally understood what eigenvalues and eigenvectors are! Thanks a lot guys, really awesome job :)

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.

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

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.

this was amazing. saved me a lot of time of struggling trying to visualize everything. may whatever you believe in bless you

This is absolutely amazing. Exactly what I was looking for, thank you.

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

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.

The first time I used eigenvectors and eigenvalues was in my continuum mechanics class. We used the symmetry of the stress tensor to find n number of eigen vectors and eigenvalues to predict crack growth in linear elastic solids. This video is definitely nice to freshen up on the theory behind eigenvectors and eigenvalues.