Singular Value Decomposition (SVD): Matrix Approximation

This video describes how the singular value decomposition (SVD) can be used for matrix approximation.
These lectures follow Chapter 1 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
Amazon: www.amazon.com/Data-Driven-Sc...
Book Website: databookuw.com
Book PDF: databookuw.com/databook.pdf
Brunton Website: eigensteve.com
This video was produced at the University of Washington

[@ayushsaraswat866]

This series is by far the best explanation of SVD that I have seen.

[@greenpumpkin172]

This channel is so underrated, your explanations and overal video presentation is really good!

[@dombowombo3076]

Don't know why you think it's underrated... Everyone who is watching this videos knows how great they are.

[@ris2043]

The best explanation of SVD. Your videos are excellent. Thank you very much!

[@AdityaDiwakarVex]

SVD was at the very end of my college LinAlg class so I never got a very good understanding of it before the final - this is truly amazing; you say "thank you" at the end of every video but it should be us saying it to you- keep doing your thing! I'm loving it.

[@smilebig3884]

The best thing about your lectures is, u do coding implementation along with huge maths.. That makes u different from rest of the traditional instructors. Kudos to you!!!

[@Eigensteve]

It's my pleasure

[@wackojacko1997]

Not an engineer/student, but I'm watching this to get a better understanding of PCA in statistics. I'm going to check the book and research this, but my only complaint (nit-picky) is trying to tell the difference when Steve speaks between "M" and "N" which I know refers to the number of rows or columns of the matrix. But really, this was great and I am thankful that this is something I can study on my own. Much appreciated.

[@omniscienceisdead8837]

you explain math in such a way as to not make someone feel stupid, but feel like their taking steps into understanding a larger concept, and the tools they need are the ones we already have, big ups

[@alexpujoldartmouth]

You have a talent for taking complicated topics and breaking them down into digestible pieces. That's the sign of a good teacher. Thank you.

[@fabou7486]

One of the best channels I have ever followed, appreciate it so much!

[@malekbaba7672]

The best explanation of SVD i have ever seen !

[@skilambi]

Please keep making these high quality lectures. They are some of the best I have seen on KZbin and that goes a long way because I watch a lot of lectures online.

[@zsun0188]

I learned this in college but couldn't recall a bit after working in the industry over a year. This explanation not only helped me refresh my memory but also enhanced my understanding as well.

[@xiaoyu5181]

This is also the best explanation of SVG I have seen! Thanks for sharing!

[@peymanzirak5400]

I find everything with these courses, even the way board arranged is just great. Many many thanks for this wonderful explanation and all your effort to make it understandable and yet complete.

[@bnglr]

every time I think it's time to pause and comment this video with "awesome", it surprises me with more informative perspective. great job

[@douglashurd4356]

Superlative production! Lighting, sound, set, rehearsals, material, these videos are among the best productions on KZbin. Even I understood some of it! :-)

[@zepingluo694]

Thank you for presenting us an amazing experience to learn about SVD!

[@sonilshrivastava1428]

One of the best videos on singular value decomposition. it not only tells the maths but also the intuition. Thanks. !

[@nathannguyen2041]

This was, by far, the most compensable explanation of what the SVD is mathematically and visually. The SVD is an incredible algorithm! Amazing how so little you could keep in order to understand the original system.

[@din_far]

this is by far the best video explaining SVD on youtube

[@rajkundaliya7796]

It doesn't get better than this. I am so thankful to you. I don't know how to repay this help.... And yes, this is a highly underrated channel

[@AkshatJha]

What a wonderful way to simplify a complicated topic such as SVD--I wish more people in academia emulated your way of teaching, Mr. Brunton.

[@NickKingIII]

Wonderful explanation, clear and easy to understand. Thank you very much

[@patf9770]

Can't overstate how good this series is...

[@nikosips]

Thank you very much for those videos , they are very explanatory . Keep up the good work, we need you lessons for our academic improvement.

[@kaiyueli1372]

This video series is so helpful!! Thank you Dr. Brunton!

[@athanasiospliousis2654]

Very, very nice explanation and presentation. Thank you!

[@Nana-wu6fb]

Literally the best svd explained, so meaningful

[@dhoomketu731]

This one's a brilliant explanation. Simply loved it.

[@HuadongWu]

the best lecture of SVD I have ever seen!

[@garrettosborne4364]

This is answering a lot of my questions on SVD.

[@alek282]

Amazing lectures, immidiately bought the book, thank you!

[@LusidDreaming]

The book is great, but relatively terse for someone like me who needs to brush up on his linear algebra. These video lectures are an excellent compliment to the book and really help drive home the concepts.

[@yasirsultani]

These are the best videos out there. Biggest fan Steve, keep it up.

[@carlossouza5151]

You are a very very gifted teacher! Thank you for sharing this! :)

[@tusharnandy6711]

Gentleman, you have done a very impressive job. I have just started exploring data science and have recently completed my college course in Linear Algebra. This was quite interesting.

[@RajeshSharma-bd5zo]

Such a cool concept of decomposition and brilliantly explained here. Big thumbs up!!

[@Multibjarne]

Explanations like this for a dummy like me makes my life so much easier

[@saitaro]

It was pleasure to watch. You should do more educational videos, mr. Brunton.

[@bryan_hiebert]

Thank you so much for posting the course material. I was running through asking ChatGPT some questions about eigenvector/eigenvalues and revisiting some linear algebra when I stumbled upon transitional probability matrices or Markov Matrices, PCA and SVD as was getting back to my Data Science studies. This is very exciting stuff and your presentation is very clear and understandable.

[@patrickgilbert6170]

Great video. Should be required viewing for anybody learning the SVD!

[@btobin86]

You are so talented at teaching, great explanations!

[@PunmasterSTP]

Matrix approximation? More like "Magnificent explanation!" I really can't convey in words how absolutely outstanding *all* of your videos are.

[@YYchen713]

Thank you for making the linear algebra less boring and really connected to data science and machine learning, this series is so much more interpretable than what my professor explains

[@PunmasterSTP]

Hey I know it's been nine months but I just came across your comment and was curious. How'd the rest of your class go?

[@eveninrose]

Just started watching this playlist, excellent explanations and a great way to promote while sharing knowledge; bought your book and can't wait to revisit w/the text!

[@Eigensteve]

Awesome, thank you!

[@deepthikiran8345]

The explanation is really wow !! Very intuitive ... thank you so much !!

[@kiaranr]

I've read about and even used SVD. But I never really understood it in this way. Thank you!

[@kansasmypie6466]

Can you do a series on QR decomposition as well? This is so useful!

[@arne9518]

This is a gold mine! thanks for your videos

[@FezanRafique]

Steve is magician of explanation.

[@Eigensteve]

Thanks so much!

[@billandrews6291]

13:41 The way I like thinking about is, for example, two vectors in R^3 that are orthogonal are not necessarily orthogonal when projected into R^2, which is essentially what is being done by dropping some of the dimensions. Love the videos though, has me thinking about SVD again.

[@juangoog]

Wow, what a wonderful presentation. Congratulations.

[@mkhex87]

To the point. Answers all the important questions. I mean you should come to the party knowing some lin alg but great for intermediate level

[@ARSHABBIR100]

Excellent explanation. Thank you very much.

[@liuhuoji]

love the video, well explained and aesthetically good.

[@Streamoon]

Thank you Prof. Brunton, excellent explanation! Just come from MIT 18.06.

[@wudiNB]

best teacher that l have ever met

[@douglasespindola5185]

Gosh, what a class! As mr. Ayush said, this was indeed by far the best SVD explanation I've seen. You've made a such complicated subject way more affordable! I wish you all the best, Steve! Greetings from Brazil!

[@Eigensteve]

Thanks so much! That is great to hear!!

[@alexyang6755]

it covers a lot.Thanks for beautiful teaching!

[@alwaysaditi2001]

Thank you so much for this easy to understand explanation. I was really struggling with the topic and this helped a lot. Thanks again 😊

[@Eigensteve]

Glad it was helpful!

[@user-hp1zj6hk5c]

really really nice explanation！you are really a great teacher！

[@guitar300k]

I like your series also the dark background make my eye feels ease than white background like other channels did

[@sanaomar2182]

This is the best explanation ever

[@parnashish1910]

Beautifully explained.

[@pilmo11]

superinformative series of SVD

[@mohiuddinshojib2647]

This is everything that I need. Thanks for nice explanation .

[@Eigensteve]

You are welcome!

[@mdmamunurrashid2945]

Love his explanation style

[@nicholashawkins1017]

Lightbulbs are finally going off when it comes to SVD cant thank you enough!

[@fabiopadovani2359]

Thank you very much. Excellent explanations.

[@khim2970]

really appreciate your efforts. wish u all the best

[@Aditya-ne4lk]

Just in time for the new semester!

[@chenqu773]

Good explanation! Many thanks ! how could one manage to get these stuffs explained in such an elegant way.

[@opencast1819]

Great lecture Steve, really enjoyed it! I have a couple of questions: is it better to have longer or shorter time series for the SVD? And is a tall skinny matrix you mentioned speaking about the "economy" SVD only matter for the memory and time savings, or is it generally recommended to have such an input matrix? Thank you in advance and best greetings from Austria) Alexander

[@inazuma3gou]

Excellent, excellent content. Thank you so much!

[@neurochannels]

I never *really* appreciated SVD until I saw this video. Mind blown!

[@andrezabona3518]

for mn ? (For example, what happen if my dataset is composed by 5000 images of 32x32?)

[@LyndaCorliss]

Top rate education, I'm happily learning a lot. Nicely done. Thank you

[@hugeride]

Just amazing explanation.

[@MassimoMorelli]

Extremely clear. Just want to point out a fact which at first did not seem obvious to me: the outer product has rank 1 because all the column are proportional to the first vector of the outer product, hence they are linearly dependent.

[@maipyaar]

Thank you for this video series.

[@mr.jizhouwubs7256]

Great video in linear space point of view. One naive question: can we make use of Lanczos algorithm such that we can pick up the most significant eigenvalues for approximation in order to circumvent the full diagonalization of the whole large matrix?

[@mohammedal-khulaifi7655]

you are at the tip-top i like your explanation

[@lekshmynair3355]

Thank you too so much sir for this explanation its truely amazing

[@maydin34]

Sir. Just thank you for making me be your student in here for free! Great performance, great job!

[@katieadamczyk937]

This is a fantastic video!!

[@jonathanschwartz7256]

Watch out Kahn Academy, Steve Brunton is coming for ya! Seriously though, these videos are fantastic :)

[@zhichaozhao172]

Prof.Steve, Thanks for your explanations. But what is the difference between the POD and SVD for aerodynamics analyeses?

[@Phi1618033]

This all sounded like gibberish until I started to think of the first term of the expansion (Sigma1*U1*V1T) as the (strongest) "signal" and the rest of the terms as ever decreasing amounts of "signal" and ever increasing amounts of "noise". So the last term (Sigmam*Um*VmT) is essentially all background "noise" in the data. Thinking of it that way, it all makes perfect sense.

[@adelheidgang8217]

incredicle explanation!

[@sachingarg4385]

Part 2 of the Eckard Young theorem is that this video is the best explanation of the theorem's part1 :P

[@regbot4432]

Wow, you are really good teacher.

[@harrypotter1155]

Mindblowing!

[@hchoudhary92]

I am trying to identify dominant modes/ coherent structures and inner-outer interaction in the turbulent wall jets using PIV images. Can you give any suggestion?

[@tobyleung96]

@14:52 No Steve, thank YOU!

[@JCatharsis]

Thank you so much professor.

[@momoh6696]

Hello once again(sorry, this will be the last I think), is there somewhere I can get some pictures like the waveform you showed several timesteps of(to be processed by SINDy i think) and the PDE of the waveform. I want to use the images with a known PDE to see if my compressed images will give something the same or similar:)

[@maciejmikulski7287]

The assumption n >> m is contrary to what we have quite often in data sciences. In many problems, the number of samples (here m) is bigger than number of features (here n). In such a case, we just take the transpose and keep going the same way? Or there are some additional considerations (of course except of swapping interpretations of eigen vectors etc)?

[@santoshshanbhogue]

Such a lucid explanation! A question though : If I do an SVD on this toy matrix X = [1, 2; 1,2], then the second singular value is zero as expected. But if I am unable to reconstruct X using sigma1*U1*V1^T, i.e. just the first singular value (The matrix is reconstructed as [1, -2; 1 -2]). But if I add an infinitesimal noise to X though, it works. Is this a well documented numerical issue?

[@franciscojavierramirezaren4722]

Thanx amaizing as always👍

[@johnberry5275]

I'm glad he made it clear that *outer products* were taking place.

[@florawoflour4501]

thank u so much sir, very helpful