Visualize Spectral Decomposition | SEE Matrix, Chapter 2
ฝัง
- เผยแพร่เมื่อ 3 มิ.ย. 2024
- 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:
1. regarding the eigenvectors of symmetric, it is correct to say the eigen vectors are orthogonal if the matrix is full rank. However, the formal definition is there always exist a orthornormal basis which contains the eigen vectors of the symmetric matrix, for details refer to • 25. Symmetric Matrices... , where professor Strang explains the case with eigen vectors with eigen value of 0.
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 :
/ 3blue1brown
and the animation software - Manim, which he wrote:
/ 3blue1brown
and the Manim Community:
docs.manim.community/en/stabl...
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!
EoP (Essence of Linear Algebra) is exactly what inspired this series. I never want to compare anything I make to the legendary 3b1b, but your comment were spot on, I hope this series can somewhat be an informal extension of of EoP. Just like in most college course linear algebra course, the first half focus on things like (vector, range, null space, determinants, linear dependence); the second half of course is almost exclusively focusing on matrix decomposition, and personally I thought the visual interpretation of them were equally valuable hence this series. Appreciate your very encouraging words, did you check out SVD by the way ?
@@visualkernel yes. It was exactly what I needed to know. I always wanted to know why all of a sudden doing the eigen decomposition of AA^T was thought of to calculate the SVD. Now I understand.
He needs to put out more videos. Viewers will come but not with just four videos. He’s got to consistently put them out.. I think he has a ton of potential.
I feel like it's because most intro linear algebra courses only cover the more basic stuff that 3b1b had already covered.
hlw swagato sir, where are you from?? guessing you're A Bengali but from where??
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! 👏👏👏
thanks buddy !
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!
Brilliant. I don't understand why the geometric interpretation is treated as if it doesn't exist in schools. Thank you a lot!
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
The best explanation I've seen in 25 years, counting from my first exposure to matrices as a freshman
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!
This is literal gold, if you are reading this comment you hit a goldmine congratulations
The music note analogy is such a beautiful way to explain the decomposition of matrix transformations! Amazing video!
Amazing, you just explained a concept in 15 minutes that my teacher failed to explain in 4 lectures. Keep being awesome!
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,
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!!
Check out the SVD one , even cooler
The decomposition of a musical chord into the notes is such a beautiful touch !!! I am lucky I got to your channel.
This video is beautifully animated and straight to the point. Thank you for illustrating decomposition in such an elegant way.
Amazing, mind blowing explanation!!!! We need more tutors like you! I wish they taught like this in high school.
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!
This is absolutely awesome, surprised you don’t get as much recognition; really wonderful video
Really good video man, I was doing the exercises like a robot just applying steps in a specie of algorithm without knowing really the meaning of it. Now I can visualize this transformations and this decompositions in my head, thank you so much
Brilliant videos that really add to the understanding and intuition of these concepts. As others have said these videos deserve far higher views, and hopefully one day you will find time to make more!
This is such an underrated video about the topic about matrix decomposition! I really like the approach that helps me visualize the meaning of the topic tremendously! Keep up the hard work!!!
THIS IS SICK! The best explanation I've ever seen on TH-cam
It was nice to see the scalings and rotations done step-by-step, very helpful for a better understanding! ty
excellent videos. I am so glad, that I found it. Thank you for a lot of work that you've done to make these excellent visualizations. Also the structure is intuitive and logically ordered. Bravo!
Really great vid!! So glad I was discovered this channel!!
This explanation was phenomenal. Great job!
One of the best tutorial videos I have ever seen. Thanks
One of the best explanations in linear algebra I have ever seen. Subscribed. Keep up the good work.
extremely well done, keep this up because college students with bad linalg professors truly appreciate the content.
Wow! This whole series is so beautiful! First time I understood so many important things!
Thanks a ton Sir. Lot of gratitude towards you for this way of knowledge!
I find this impressive, this concepts are really hard to understand but the order and the way they are explained made very much sense to me! Thank you!
Brilliant! I wish professors emphasized this intuitive understanding more often
love from India, 3blue1brown and you guys really made algebra beautiful; again.
Very nice presentation. Like a great athlete, you make what you do seem effortless.
thank you sooo much for these videos. for some reason I remember/understand concepts so much easier when they have visual interpretations
This is amazing, thank you for all your high quality videos! Can't believe this has comparatively few views
Dude I think this might be my first comment on youtube, but you deserve it haha. This series got me understanding matrices at a level even deeper than youtube-linalg's godfather 3blue1brown:) You ROCK! and dope logo you got for your channel! keep it up, I'll be there;)
really loved this video!!!! thank you so much!!! made the concept so much clearer!
The logic flow renders much more understanding than other 'visual course', particularly about the eigenvector!!!
Most clear and illuminating
Thank you, this was beautiful and very informative! In my opnion, getting the intuition behind a concept is the best way to remember it and to apply it in a different context
This is an amazing production. Thanks
Insane visualisations!
Helped me in the understanding of diagonalisation of matrices immensely
Thanks a lot!!
Thanks for these kind of content, I feel by having these kind of content you are helping people to understand power of mathematics and help apply these concepts which otherwise need tedious study of mathematical formulas , which helps people like us to bridge the gap to understand other applied solutions from these basics. Keep it up , world is converging and future solutions can only be solved when more people understand the basics intuitively and apply to solve new problems of world
So happy to stumble on these, you got a new subscriber today!
Watched this video again, and oh boy its so good!
thank you so much for introducing me to this. You've made my life easier through your work.
Great extension to 3b1b's Linear Algebra series, Thanks so much !!! it really help me build the intuition
This video was infinitely awesome!
This is great! I took a course in Operator Theory and for the life of me couldn't visualize the spectral theorem in any intuitive way. I wish I found this video months ago!
Holy crap, the Manim animations are incredible. I want to use Manim for my own channel for unrelated to math topics, and I am going to take inspiration from your style. It's probably the best I've seen aside from 3B1b.
The guitar chord analogy was a nice touch
Please keep doing this masterpieces! Thank you!
Neo would be proud of these vids. Fantastic job!
Amazing videos. So underrated!!
Thank you so much for this video it really helped me better understand the concept!
Excellent video! Would love to see more content like this :)
these videos are amazing，Thank you for sharing this with us.
Very clear transparent video. It has also powerful mesmerizing effect. Calming satisfying Yoga-effect😅
Fantastic work, thank you for this
So good!!!! Especially your explanation of eigen stuff.
Thank you for this video. Thank you very much. It's awesome. I want more of your videos.
Amazing explanation. Keep up this good work
Amazing. I am stunned. Thank you.
Beautiful video, i was amazed about the quality and how good the explanations and vizualisations were!
I dont see the chapter 1 part anywhere, is there only supposed to be the chapter 2?
love this videos...they´re so helpful......thanks man.....keep doing it......by the way, the chords detail was pretty nice....
really supreme video! thanks a lot!
very helpful. thank you. never expect math can be so easy to understand.
3b1b should shout this out man. Legend
A pretty much under-appreciated course on LA
Thank you so much for your help!!!
Video is transformative
this is fantastic!
Thank you so much.
This actually complements some of the illustrations 3B1B didn't demonstrate.
I love these videos! I've watched 3Blue1Brown linear algebra videos. They are good but more conceptual math. These videos show how to implement linear algebra to computer graphics. I'm lucky to find it when I search svg
Yo this vid might of just saved for my finals thx düd
Keep up great work man, I hope you can release more😢
Beautiful!
Thank you so much sir :)
Very nice, thank you very much:)
Thank you very much!!! Hope you will find the time to make more video's:) You helped me a lot!:)
It is interesting how I make this kind of video while expecting to be seen by no one. But youtube recommendation algorithm just does interesting things, leading people to the edge of the internet. And I am very glad if it could have offered an alternative perspective on how we view matrix decomposition. SVD coming out soon, that will be a good one.
@@visualkernel
Why expecting to see by no one ? I personally really like math videos and these are really great and definitely give good fight for 3blue1brown
@@omridrori3286 Well, this video is currently not advertised, and finding a video like this is pretty challenging. (I actually tried to TH-cam search "Visualize Spectral Decomposition" explicitly, and my video does not even come out ). So I think it's very interesting how you landed on this piece of recommendation. And about 3b1b ... I think everyone who makes math video can only asymptotically approach the half of what he is. Thank you Omri, for your words of encouragement.
@@visualkernel
When i really wait for that every day i check please i really want to know svd
Brilliant! 👏👏👏
This has gotten me to see the linear transformations of these "sandwich" estimators. I wonder how we can visualize econometric estimators like the basic OLS (X'X)^-1X'Y and it's covariance matrix (X'X)^-1(X'e')(eX)(X'X)^-1 and even longer matrix multiplication models.
Great video
Please, continue this series. Make about PCA !!!
Respect for you
Bro just keep making these vidoes and the views will eventually come, it is literally inevitable!
brilliant!!
Im feeling kinda sad because i can only found these amazing visually explained videos when i have project at uni.
Excellent
Brilliant!
Thanks
Nice video mate just wanted to know how did you get the conceptual knowledge
Amazing video, fantastic job. Small mistake - at 14:50, the labels for Q and L are incorrectly swapped - L is labeled as the eigenvector matrix and Q is labeled as the eigenvalue matrix, but the reverse is correct.
yes . Eigenvalues scale. Eigenvectors rotate.
sublime
can you say which software you using to visualize, By the way lecture was amazing
Brilliant video. I have a question - at 12.29 , you say Eigen rotates to align with Standard Basis while just earlier you said the opposite for Q - Standard basis rotates to align with Eigen. I am confused....
Yes. That's confusing.
Visual Kernel, do I understand correctly that Q.T gets us from standard to eigen basis whereas Q gets us back to standard basis?
3:11 , can't stop laughing from the 3B1B reference of pi...😂
Thx for the video! Couple of questions:
1. Does the SD(spectral decomposition) method, work only for full rank symmetric matrices? What are in general the conditions for the SD to work?
2. Would this work also for complex-valued symmetric matrices?
2. Yes. Complex valued symmetric matrix is called Hermitian matrix. You can perform similarity transformation with the help unitary matrix. Which are complex analogue of orthogonal matrix.
Symmetric matrix S => Hermitian matrix H
Orthogonal matrix O => Unitary matrix U
You have O'SO similar for complex case
You get (U*)'HU
Here * represents complex conjugation and ' represents trasponse. There is single notation to represent it called dagger. It is not available in my keyboard so I broke it down.
You have to take complex conjugation and then transport it.