Visualize Spectral Decomposition | SEE Matrix, Chapter 2

A video illustrating the underlying elegant visual interpretation of Spectral Decomposition.

Chapters:
0:00 Chapter 1 Summary
1:23 Symmetric Matrix ?
1:44 Property of Transpose
3:27 Matrix Decomposition
4:58 Eigen Vectors and Eigen Values
8:07 Strong Property of Symmetric Matrix
9:36 Spectral Decomposition
12:03 Visualization
14:13 appreciation

Video Sins:

2. 00:50: a diagonal matrix is also a symmetric matrix, and its visual interpretation is very straightforward. So it’s definitely not correct to say every symmetric matrix produces a visually complicated transformation. But normally when we address a matrix as a symmetric, we often mean “symmetric and non-diagonal”, if the matrix is diagonal, we would just call it diagonal matrix.
3. 2:46, the transpose of orthogonal matrix is also a orthogonal matrix. I didn’t mention this fact.
4. 7:42, when the eigen value is 0, it means the vector is in the null-space, pretty important fact, which I didn’t mention.
5. 14:20: the visual interpretation of matrix decomposition doesn’t apply to every type of matrix decomposition. Most matrix decomposition are often used for speeding up computational purposes, for example the LU decomposition.
6. Order of eigen vector could be swap, so there are many alternative decomposition of symmetric matrices.

This video wouldn’t be possible without the inspiration of the legendary 3b1b :

and the animation software - Manim, which he wrote:

and the Manim Community:

[swagatochatterjee]

I literally have zero ideas, why does this series have so less views. This is literally the best extension to 3b1b's Linear Algebra series. This is exceptionally exquisite! Subbed!

[hansenmarc]

I think 3B1B would be proud. This provides very nice intuition into how to think about spectral decomposition and the role of symmetric and orthogonal matrices in it. Well done, sir! 👏👏👏

[supdawg]

My god, no one in the math community intuitively spoon feeds me this stuff, but that's what I need. And these videos do exactly that. This is fantastic. I truly believe that the people that substantially innovate in any field are the ones that understand that actual intuition.

[darshan]

Brilliant. I don't understand why the geometric interpretation is treated as if it doesn't exist in schools. Thank you a lot!

[aryanpinto]

best teacher ever. I´ve never lernt a tough consept this fast + the background music is the most fitting thing.

[Automan]

God fucking damn bro. Even my two brain wrinkles managed to understand the intuition behind decomposition. The most elegant explanation I've ever seen of the subject, and the lead up to it with all the topics was masterful

[leticiabarbaneramenezes]

this series has just become the best videos I've EVER watched in my life! Never thought I'd cry in the middle of my Uni's library bc i got emotional on how spectacular and genuinely beautiful linear algebra is!

[lakshyadugar]

Probably the best video I've come across explaining this. This not only made SVD crystal clear but also cleared so many other concepts within Linear Algebra not even directly related to SVD. So glad to have come across this. Looks like you haven't uploaded in a couple of years, hoping for new videos!

[Colby_ye]

The music note analogy is such a beautiful way to explain the decomposition of matrix transformations! Amazing video!

[willwu]

Damn this is so intuitive! Its easy to get lost in matrix multiplication but this was a superb reminder for me to just step back and just think of them as geometric transformations. Its actually kind of earth shattering. TY!!

[dimitrismylonas]

This video is beautifully animated and straight to the point. Thank you for illustrating decomposition in such an elegant way.

[nguyenquannnn]

Amazing, you just explained a concept in 15 minutes that my teacher failed to explain in 4 lectures. Keep being awesome!

[JunZhang-ralphjzhang]

The best explanation I've seen in 25 years, counting from my first exposure to matrices as a freshman

[max]

This may not mean much from a stranger on the internet, but I don't know if I can convey with words how amazing and effective this video is. This is a masterpiece, and it stands far above the numerous math content I've seen (believe me, I've seen a lot). You have a real and rare skill for being extraordinarily concise and intuitive. Keep it up, <3

[kaansun]

This is literal gold, if you are reading this comment you hit a goldmine congratulations

[nicolasmanelli]

Amazing to be honest. One of the rare channel to explain the intuitions rather than just the computations. Bravo !

[md.salahuddinparvez]

The decomposition of a musical chord into the notes is such a beautiful touch !!! I am lucky I got to your channel.

[karlrombauts]

It took some digging, but I am so glad that I found your videos. These are absolutely fantastic! I really hope that you get more subscribers and that you continue producing content. These videos have been exactly what I needed. Amazing work!

[TMK]

This is absolutely breath-taking!

WOW!!!

[LydellAaron]

Very well done! You've done a very good job and now (meaning I've been around and around, many videos on this topic and advance mathematics) I would recommend people watch yours first because you lay out all the ideas and connect them, and the visual geometries behind them.