Motivating Eigenvalues and Eigenvectors with Differential Equations

This video *derives* the eigenvalues and eigenvectors of a matrix "A" to solve the matrix system of equations dx/dt = A*x.

@eigensteve on Twitter

This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Overview
1:26 Problem Setup
2:33 The Simplest System of ODEs: Decoupled (Diagonal) Dynamics
10:25 Finding a Coordinate System Where Dynamics Look Diagonal
15:42 Deriving the Eigenvalue/Eigenvector Equation
19:03 A Picture of the Eigenvalue Equation
22:27 Recap and Summary

[aedin]

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.

[hydropage]

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

[edzion]

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

[aen]

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

[zjason]

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.

[saraiva]

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

[guang-yuding]

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

[aramesh]

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.

[jimlbeaver]

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

[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...

[umedina]

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

[anaslahrichi]

Best explanation I ever heard about eigenvalues

[Leopoldinow]

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

[mintakan]

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.

[agrajyadav]

You are such a wonderful teacher, Sir!

[hoseinzahedifar]

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

[wesleymesquita]

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!

[Alliban]

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.

[dennislui]

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!!!

[star-uyjc]

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.