What eigenvalues and eigenvectors mean geometrically 

Finally understood the essence of eigen values; some people are really meant to teach math and others are not. Thanks Dr Bazett

Finally!! A short and sweet video on eigen values and eigen vectors. Thank you for explaining it so well!

Fantastic! You have a gift as a teacher.

probably eigen problem is such a topic which , in many textbooks is not described geometrically. Rather it is described analytically and does mot make a clear sense. But here you have given it's geometrical description which is very important for beginners like me. Thank you very much sir.

I love the simplicity and the words just synchronised with my brain. Such a beautiful explanation. Thank You Dr Bazett

This is the BEST video I have watched on this topic. Helped a lot understanding how PCA works. Thanks so much!

I rarely comment, but this was an amazing explanation, it really helped me to understand and visualize these concepts! Thank you!

Thank you!! You explained the essence of eigenvalues and eigenvectors very clearly! You are an amazing teacher!

Wow! Thank you for this! You have no idea how many people I've asked this question and no one could give me an answer! As a self-learner, I finally get this important concept! THANK YOU!

Hey Trefor Bazett, I just wanted to thank you and share my gratitude with you for making this series. This past month I've taken Linear Algebra and C++ simultaneously, while also holding down a part-time job working QA at the SMU Physics Dept; so as you might be imagining, I didn't quite have the luxury of free time to spare. I ended up reviewing for both finals this past weekend, but still struggled to fully understand how everything connected together: I honestly felt like I was completely screwed. However, by way of happenstance, I googled, "Visualizating Diagonalization and Eigenbasis", and came across your video lecturing on it. After watching it and still not grasping the related concepts, I decided to start watching your playlist from episode 12 onward (keep in mind this was at 3AM this morning--my final was at 1PM). I opted to wait and look at my final review after I completed your 81st video that way I could make the most out of my time and go into it knowing what I needed to know. Well, friend...I gotta say, you basically taught me the basic introduction to Linear Algebra in less than 10 hours (I watched most of your videos at 1.5-2x speed with the most of the final videos at 1.25-1.5x speed). Just a few hours ago I came out of my exam, and I finally received my results........... 85%!!!! Now, I'm slightly disappointed I messed up on a few silly mistakes I made, but dang, man, I have to say I COULD NOT HAVE ACCOMPLISHED THIS WITHOUT YOU. Thank you, sir, for all the time and effort and attention to detail and specific example problems that you put into your videos. If anyone else is reading this after watching this whole playlist, congratulations! You now have a pretty solid grasp of the concepts of Linear Algebra. If you're reading this, but have not watched this many of Trefor's videos on Lin. Alg. and are struggling to grasp the concepts, DON'T WORRY! There's a UNIQUE, yet TRIVIAL Soln. (; to your problem (ha..I'm punny......#LinearAlgebraJokes); all you need to do is sit down and watch through the majority (if not all) of the videos in his playlist. If you're short on time, that's still no excuse not to go through it. I LITERALLY did it in LESS THAN HALF A DAY, and still had time to eat and review other material. That means you can/should too; you won't regret it. I'd love to post my two exam reviews along with the blank copies of the two actual exams I took this month, all of which include solutions with explanations of course; however, I'm not sure if I can do that here. If I can't, or you can't seem to find it please don't hesitate to DM me. I'll be more than happy to provide that. (I'm a programmer, so I'm almost always on my computer, and if I coincidentally am not, I most certainly have it nearby, so you can expect a pretty rapid response.) Heck, maybe I just might send it to you, Trefor--that way you can do another couple videos going over them!! That'd be awesome. I'd actually watch you go through them again.

Such a great video! This is the first time eigenvalues and eigenvectors REALLY clicked visually in my head. Thank you! :)

by far the best education video to understand this tricky concept , thank you

wow !! i have already done course on linear algebra but didn't knew what they do. indeed a great video

It's always tricky to figure out how best to explain eigenvectors to students who are "new" to Linear Algebra. This video is great!

Such a simple and great explanation. You really simplified the whole eigen problem. Thanks you sir, you are a great asset of humanity.

I am studying Neural Networks and Deep Learning course here at my University. I was finding hard time at understanding eigen values and eigenvectors. This gave me a crystal clear understanding of what eigenvalues and eigenvectors are. Thank you so much!

Finally, I made sense of a lesson talking about eigenvalues and eigenvectors! Thanks! Really appreciate it! You're amazing - to say the least!

Such an amazing Geometrical Interpretation for Eigen Values and Eigen Vectors. This helped widen the scope for thinking what exactly an eigen value and eigen vector relate to a matrix. Thankyou for the explanation

Such a good explanation! I wish they explained it more geometrically in my lectures,’

Very nicely explained! After several years, have understood eigen values and eigen vectors finally.

Sir you each lecture is full of intuition.. but this lecture is much more intuitive.

This is fantastic. This series is teaching the entire course in the course of a day or two. My university had chosen to introduce matrix operations prior to L.T; each component building rather than providing and, moreover, allowing the clarity of a larger, interconnected picture. Hence guessing and frequently misunderstanding the point of the professor, leading to a hazier understanding of the topic post course than prior. Lots of connections made here, and enlightenment on the point and meaning of things. Straight to the point. Love it! Even more so then 3B1B’s videos, I personally find that these allow more so for visualization/personal visualization, which I find key-critical. The perfect guidance towards that, and incredibly engaging and clear→ straightforward and logical, assistive steps and guidance → explanation. Fantastic and a gem of a discovery, would love to be a student here!

Wow, this now makes intuitive sense. Excellent.

What an explanation Sir, loved it :) Thanks for sharing :)

This was absolutely helpful! Got a very clear picture. Thanks a ton!

Learned and forgot about eigenvalues and eigenvectors a couple of times and never understood what they intuitively mean and why they are useful. Just want to say thankyou for your intuitive explanation.

This immediately helped my understanding of it: from basically, for all intents and purposes, 0 to 100 regarding the contents presented herein.

really thanks for explaining the "why" in this complex world. Most ppl start by explaining the how.

this is flippin' awesome! I've been using eigen concepts, in some citizen science projects, for a bit now but could only view it algebraically. Linear Algebra in college focused on the algebraic interpretations. This finally helped me visualize the geometric interpretation!

This was an amazing explanation, even though my college has taught this, this video actually helped me understand it in a better way. Thank you

I have watched enough Linear Algebra videos over the course of this semester, that this time, I have hit "full Engineer face"

Ive had 3 classes that talked eigenvalues and eigenvectors, but this video explained it best :D

Nice explanation very concise right from basics. Perhaps a small slip at 3.55 says that we are just looking for just a STRETCH of a vector not any rotation, inversion, projection or shear... but an inversion is kind of okay if we take it as a stretch with scale or magnification -1 which is kind of needed for negative eigenvalues. Hope this helps

Only when you really understand the subject you can give these kind of intuitive examples. Nature has been very kind for you, Trefor. And I think you made the most of it. Chapeau! P.S. Don't tell me you can play piano and violin too ;-)

Thank you Dr. Bazett, your explanation open up my understanding on this topic.

beautiful concise general explanation

wonderful! Thanks for such a nice and subtle explanation.

Clear and streight to the point explanation, thanks

Thanks ...... great explanation at Ax = λx ..

You saved me... Thanks a lot... Now my concept is very clear...... Many many thanks...

Quite good video, awesome!

Thank you! that was helpful and understandable

So we are just going to ignore the fact that this is the honeybadger documentary voice giving a kickass explanation of eigenvalues?

Excellent way of explanation, Sir, Keep it up!

your animation and explanation are the best part!!!

Lovely explanation.

Fantastic ,Finally understood the meaning of eigen values

This is exactly what I was looking for

well explained, truly helped me to understand thanks a lot. Please do more videos like this for jacobian matrix, Lagrange multipliers,

Simply beautiful. I knew the mathematics behind this but the geometrical representation was very informative. I frequently use the eigen system in fea - modal analysis. So this is quite helpful.

beautifully explained

Love and respect from INDIA🇮🇳🇮🇳🇮🇳

Thank you. This video helps a lot in my end. Though I have to replay some parts over and over again due to fast discussion and foreign accent.

So nicely explained 👌

Indeed, a great explanation.

Excellent. It makes sense (after all). Thanks

Awesome Explanation Sir!!

Best on RU-vid

Thank you sooo much. I finally find out what eigenvalue actually means besides knowing how to calculate it🎉

Really good explanation! Thank you Sir

I vote we replace professors at universities with such youtube videos. Excellent explanation of the actual purpose behind this concept.

Worth mentioning that the eigenvalues(stretch factors) 2 and 3 of the matrix show up on the leading diagonal elements (sort of a happy accident) which ONLY happens for a triangular matrix, as here (all zeros on one side of the lead diagonal)

Muy buena explicación, simple e intuitiva, ojalá me lo hubieran explicado así de fácil hace casi 30 años

AWESOME !!!! Thanks for the video!!

Fantastic explanation

further to Trefor idea that the zero vector is silly as an eigenvector we might also note that the zero vector doesnt really point in any well defined or specific direction and it couldnt be magnified out to infinity like usual eigen vectors so i think it can be excluded based on logic as well as just being silly:)

Thank you, really helped me understand

Thanks a lot for explaining it's meaning

Thank you Sir for the nice explanation ..

This video is awesome. Thank you!!

subbed for that enthusiasm!

Incredible video. thank you!

Eigen vectors are those which shuts the matrix. They are like come mess with me, all you do is mess with my length, you do not mess with my direction!

Damn, this is crazily simplified here

Amazing video

Eigenvalues are stretching factors? Thank you for your tips!

How when you multiplied e1 by A it rotated, it should be stretched along x axis since y=0

great explanation!

wow solid explanation, dear sir

More than splendid

Wow What a clarity

Really it helped me! Thanks :)

A two minute lecture crammed into 9 minutes. So what if you compute the eigenvalues and eigenvactors for some matrix A. What do you use them for? What is an application for the eigenvectors?

How about discussing a practical application of eigenvalues such as in facial recognition algorithms.

you saved my life!! many thanks

Nice video! You earn one more subscriber

Awesome video. Liked and commented to please the algorithm hehe

Cool explanation!

great explanation

So it's fair to say that the eigenvectors express an alternativ set of dimensions that span a space which is not screwed by the matrix - thereby spanning a space that can be expressed as strictly and simply linear ?

thank youuu!!!!!!!!!

Wow so simple 🤩

Does this mean that Eigenvectors are always for a particular transformation, for a different transformation the same Eigenvector may not be a Eigenvector?

Well explained

This is great, thanks

great video

incredibly useful

We can find Eigen values to matrix by solving it's characterisitc polynomial equation. How do we find Eigen vectors for any matrix A

It's just wow😱😱😱

thanks