Eigenvectors and eigenvalues | Chapter 14, Essence of linear algebra

A visual understanding of eigenvectors, eigenvalues, and the usefulness of an eigenbasis.
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Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.

A solution to the puzzle at the end:

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

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3blue1brown is a channel about animating math, in all senses of the word animate.

Various social media stuffs:

[Thien--Nguyen]

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!

[madelinescyphers]

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.

[VainCape]

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

[Cena]

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.

[thenumber]

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 :)

[RealEngineering]

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

[michaelkotthaus]

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".

[zhuolovesmath]

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

[chadmaster]

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

[tomtom]

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.

[Emoechaiti]

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

[meghmalharmanekar]

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.

[Robert-jyjm]

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

[ProfessorEisenoxid]

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

[Buckybarnesfan]

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!

[marianseidl]

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!

[HAAH]

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

[alyoco]

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

[techiesithastobetechies.]

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!

[AntonKimS]

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