Motivating Eigenvalues and Eigenvectors with Differential Equations

[@aedin6397]

Going through this material in school the first time was like driving a jalopy down a bumpy dirt road. Seeing Professor Brunton's crystal clear explanations of the same material now is like driving a Cadillac down a newly paved 4 lane highway. These presentations are invaluable and I envy students today who have access to Dr. Brunton's extraordinary library of lectures.

[@hydropage2855]

A really beautiful insight I was told is that eigenthings turn matrix multiplication, into scalar multiplication

[@edzion6433]

Professor, you're a fantastic teacher! This was a fantastic lecture.

[@guang-yuding3778]

Really illuminating lecture helping me to understand why eigenvalues and eigenvectors are so important and frequently mentioned in this field! Thank you, Professor Brunton!

[@aramesh428]

This video couldn't have come at a better time. We're covering this exact topic right now in my Controls Class. Thank you Dr. Brunton.

[@pavansinghyadav9975]

exactlyu same here I have my presentation on Dynamic mode decomposition and this video Is helpful in a lot way

[@zjason7582]

This video lectures are outstanding not mention this is free to the public. For me, this is useful to help me understand the auto-control theory in the perspective of math. Thanks so much Professor Steve. Respect from China.

[@Eigensteve]

Thank you! Glad they are helpful :)

[@AbidAli-bv2gl]

Great teacher, excellent video

[@mintakan003]

Best explanation I've heard, on motivating eigenvalue and eigenvectors. When I was first introduced to it, it was just an equation, to define what it is (A*V = L*V). Yeah, there was the mention of basis vectors, and diagonal matrix. But the motivation is to simplify things, keep things clean. The zoo metaphor is a good one. Does one want everything nicely organized, dealing with one animal at a time, or is it the whole confusing mess, with everything interacting with everything else, out in the wild.

[@umedina98]

Thanks for sharing this amazing lecture! The zoo/jungle analogy was simply perfect, and that last board snap is worth gold!

[@arthurcpiazzi]

Amazing, as usual

[@anaslahrichi]

Best explanation I ever heard about eigenvalues

[@Leopoldinow]

Flawless explanation and motivation. Just the best lecture on this topic! Thank you for sharing.

[@agrajyadav2951]

You are such a wonderful teacher, Sir!

[@hoseinzahedifar1562]

Great great lecture...Easy to understand... Thank you very much...❤❤❤

[@Alliban59]

When I first heard about "eigenvalues" (in a boring mathematics lecture), I thought: "What the hell is this?" When it came to their application, I have begun to love them.

[@aik21899]

Is there a behind the scenes on these videos? How are they shot?

[@wentaowu3070]

This lecture is really amazing

[@star-uy3jc]

Thank you very much! Can you please tell us about the concept of rank and in particular why we are interested in low rank solutions.

[@dennislui2938]

Steve, I'm working late and need something to help me focus. Should I play Beethoven's No. 9? No. Eigen_steve talk about my favorite topic eigenvalues? Yes!!!

[@enisten]

Steve is the GOAT 🐐🙏

[@willson8246]

This video is really really cool! I've ever found the equation A*V = Lambda*V (The same one as the last equation you wrote) from some text book but not found the Eigenvalue system equation (which contains multiple A*V = Lambda*V equations) like this. Could you please suggest/share which text book I can find this Eigenvalue system equation?

[@saraiva407]

I didn't take such advanced DE courses in my undergrad, but now I kind of feel like I did!! You're the best!! :D

[@zombieshoe]

This motivation for eigenvalues/eigenvectors really helps me understand the value of the decomposition!

[@ΚωνσταντίνοςΛαζαρίδης-ξ9ι]

Thank you sir!

[@ANTGPRO]

Great ❤

[@baozhao1870]

Hello Professor, it's really a nice video! I have a small question here, sometimes we have an ill-conditioned system matrix A, and when we do the eigenvalue decomposition, we will inevitably meet small and large eigenvalues (eigenmodes). This kind of system may sensitive to small errors during time evolution. Do you think it's a good idea to do some mode truncations to neglect some fast dynamics in the system?

[@adrianom]

Well, it depends on what you are interested in. That's what is done in dominant pole approximation for example (large negative eigenvalues, i.e. fast dynamics, are neglected)

[@et4493]

No thank you Sir! Outstanding lecture

[@-slt]

Beautiful. absolutly beautiful. 👌👌👌🔥

[@star-uy3jc]

What is the relation to svd then?

[@KHIRD]

Thanks

[@graystone2802]

I’ve been wondering, how does Steve write on the board? Is he writing backwards so that we see the mirror image?

[@rjScubaSki]

He’s writing on a transparent board, then the image is flipped left to right.

[@lioneloddo]

A such relationship between the diagonalization of a matrix and the specific tone of a guitar string is something that has alway amazed me! How is it possible that just by transforming a matrix, we can describe the reality? Nature is not a mathematician who calculates the eigen values and eigen vectors of a huge matrix! I don't understand what is this mysterious link between matrix computation and the physical world...

[@mpoullet]

For a more mathematical approach to this topic, I'd recommend "L. Sadun, Applied linear algebra. The decoupling principle. 2nd ed. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1140.15001)". The videos for this book are also on KZbin: kzbin.info/aero/PLZcI2rZdDGQrb4VjOoMm2-o7Fu_mvij8F

[@khayahbrookes]

Hilbert's Zoo. smh. Thank you.

[@aen0108]

Best explanation to come from a system of diff. eq´s. to a eigenvalue problem! well done!

[@sinaasadiyan]

Great

[@jimlbeaver]

I had never heard of this motivation and never put it together myself. Excellent, I always wondered why it worked! Thanks.

[@curtpiazza1688]

This is great!....Linear Algebra and DEs ! ❤ 😂

[@kottybeats]

Incredible lecture!!

[@alxjones]

I like the idea, but it sort of misses the full scope of what eigenvalues and eigenvectors are. Indeed, if you have a system of distinct eigenvalues which are all present in your ground field, then you've got the result here. However, the fact diagonalization is not always the case despite the existence of eigenvalues and eigenvectors means that defining eigenvectors in terms of diagonalization doesn't actually make mathematical sense. It's a great motivation, but I'd expect to see the other cases handled in a similarly geometric fashion to show exactly what those cases entail.

[@adaisya]

Thank you for the amazing lectures!! I was wondering if there's a way to access the homework solutions so that I can check my answers? The links for them on the course website don't seem to work.

[@tupublicoful]

If you are an engineering student and getting a bit frustrated with differential equations please checkout this playlist. It’s powerful and beautiful content.

[@wesleymesquita8380]

Wow! That is a mind blowing class. Thank you! It is always a challenge to link topics in math, but when done it is beautiful. I would say everytime some piece of math is teached it should come up along some motivation, why that tool will be needed somewhere in the future. I remember how useless was to invert matrices in high school or learning eigen-stuff on second semester of college. It is satisfying getting things together!