Eigenvalues and Eigenvectors

This video explores the eigenvalues and eigenvectors of a matrix "A". This is one of the most important concepts in linear algebra. The eigenvectors represent a change of coordinates in which the "A" matrix becomes diagonal, with entries given by the eigenvalues. This allows us to easily solve arbitrarily complex linear systems of differential equations.

@eigensteve on Twitter

This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Overview and Eigenvalue Equation
3:15 Eigenvalues and Eigenvectors are "Special"
6:00 Example 2x2 Matrix
14:55 Computing Eigenvalues and Eigenvectors for *any* Matrix
21:18 The Determinant Measures Area of a Transformation
25:50 Determinant of 3x3 Matrix
28:35 Revisit 2x2 Matrix Example

[dennislui]

"Not a perfectionist. But I do have standards." -- eigensteve 🤣🤣🤣

[TNTsundar]

This is such an Eigen lecture! Thank you for teaching us!

[pierricbross]

Just want you to know these videos are great. If you have any self doubt about anything like the time you spend thinking or whatever it makes the video better, not worse. Love the time you spend making sure everything is legible and coloured, and the explanations. You'll be out-earning Organic Chemistry Tutor in no time.

[patriksandahl]

By following your videos, and by carefully working out all the examples myself I finally understand things that I've been struggling with for years. Your IRL students are previleged to have you as their professor.

[giovanniminelli]

Love that graphical approach in the cartesian plane. Those were concepts usually taught as abstract properties but having an image in my mind helped me a lot!! Thank you!

[hoseinzahedifar]

Your method of teaching these mathematical concepts is amazing... Thank you.❤❤❤

[potter-otter]

"Not a perfectionist but I do have standards" love it! 😂

[hasinabrar]

I have completed my masters in mechatronics and automatic control recently and a huge portion of my knowledge is from your videos and from your book data driven engineering. If I ever do a PhD in United States, It would be an honour for me to perhaps work with you or be under your wing somehow.

[anirbanchel]

Thank u sir for doing this.... I live in a third world country and never been taught linear algebra the way u did...

[press]

I took controls 30yrs ago in engineering (chemical) from Prof MF Doherty, time-series, stochastics, control expert. Excellent class, excellent prof. The BEST part, which I'm patiently waiting for you to get into, was closed loop control, mostly in freq-domain (s-transforms). PID control design, moving poles and zeroes, control figures-of-merit. Super stuff, brings e-values/e-vectors and eng math to real life. Controllers (PID for sure) are everywhere.

I also took a course in z-transforms, digital process control. I wonder, does nobody use z-transforms anymore? Is that math dead, replaced by computation? I've never seen/heard of z-transform again. Kind of like linear-programming: I took one course, and never heard of it again.

ps: I hope your YT vids boost your tenure status. It's a fine piece of work you're collecting. Especially love that black board you got. There's a physics prof on YT who uses same (ouch, I forget his name). Impressive piece of kit.

[StaticMusic]

I love this, eigenSteve! Incredibly grateful for your lectures and the huge effort you put in

[rajendramisir]

Brilliant lecture Professor Bruton. I really enjoyed it. Interesting for me to learn that eigen vectors and eigen values are used to solve ordinary differential equations. Just fascinating stuff!

[niz_T]

Finally i got a clear understanding of the topic - thanks a lot!! 👍

[wolfisr]

Dear Prof Brunton, thanks for this detailed series - i was never taught the deep connection between lin algebra and diff equations like you do it here.
I think it is worth mentioning that all the diff equations you use here (in all the chapters of the series up till here) are not only linear but also HOMOGENEOUES. Thank!

[rasjon]

Great lecture! Another cool insight: the A matrix from the example is a symmetric matrix. the eigenvectors of a symmetric matrix are always orthogonal, as one can also see in the drawn coordinate system. this is important whenever A is a covariance matrix (a covariance matrix is always symmetric), e.g. for the PCA this means that the PCs are always orthogonal

[GabriellaVLara]

why did no one teach us like this at the University? :') thank you for an amazing video!

[danielsmb]

Prof Brunton, I knew something deep inside the Eigen (historically, linguistically and technical) many years ago. I studied it almost every day for about 20 years and still don’t understand fully. Due to the symmetry of the birthdate, I named my son Eigen. My translation on Eigen is fundamental law necessary to keep equilibrium under dynamics propagation. Hope to see more video on Eigen in explaining the universe, in chaos, there is a cosmos; in all disorders, a secret orders (Carl Jung) ❤

[mauYair]

It seems to me that "eigen" means self/own. I imagine the unmovable directions associated with the transformation, all directions end up somewhere else, the eigenvectors are the transformation's own (self) directions. I speak Portuguese and Hebrew, in both languages eigen is translated as self: autovetor/autovalor (pt); ערך עצמי / וקטור עצמי (he).
Awesome videos, Steve! you rock!

[alexcook]

Perfect for soft iron compass calibration

[jones]

Thanks for the memories. I took linear algebra a lifetime ago. It was one semester if memory serves, and we got as far as eigen vectors and values. At the time I had no idea what was the goal of it all. Except how to find the values and vectors. I was in the tall weeds.
So, diff equations, huh?