Matrix Factorization - Numberphile
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- Опубліковано 15 тра 2020
- Featuring Professor David Eisenbud, director of the Mathematical Sciences Research Institute (MSRI).
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More videos with Professor Eisenbud: bit.ly/Eisenbud_Videos
More form the Professor on our podcast: • A Proof in the Drawer ...
The 17-gon: • The Amazing Heptadecag...
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Dr. Eisenbud seems like such a nice guy
He *is* a nice guy.
@@numberphile Why do you seem serious?
He absolutely is!
Totally
"DR. EISENBUD IS INDEED A HUMAN WHO IS NICE" *blinks "HELP" in morse code*
According to Google Scholar, "Homological algebra on a complete intersection, with an application to group representations" has 678 citations.
huh
I feel like we should also be including papers that cite those 678 papers, and so forth, if we are using citations to measure impact.
@@PHDnHorribleness "What is the cardinality of the set Q, where Q is the set of all papers that either cite "Homological algebra on a complete intersection, with an application to group representations" or cite a paper in set Q"
@@CommodoreHorrible You should write a paper on it and then do the calculations on your own paper.
For a pure mathematics paper, that's a lot. In statistics, medicine, etc. you get different orders of magnitude, but there's less honesty in those numbers. In math, for example, you would typically only cite papers that are directly relevant to what you're doing (just as you would put authors in alphabetical order and don't include coauthors unless they contributed).
It's pretty neat how he basically did "market research" on the physicists to see what paper they might like next, like the next version of a product. I've never thought about research fields interacting in that way.
no i think it went the other way. He wrote the paper first and then the physicists found it useful and it became popular.
@@bonob0123 David Vaughan was talking about the generalization paper, not the initial paper.
i know right?
@@Isiloron Fair enough
@@Isiloron Thing is the physicists are usually like "I don't need the generalised version I just need enough to solve this specific problem." Then a hundred years later they come back with a "what were you saying about the n-dimensional generalisation again?"
love his answer at 15:01 "It makes me pleased, that's all really." :)
@@ryanhenrydean1584 Thanks for pointing out his Username
12:23
"So the reason that x was ok here is because it was multiplied by..."
"...zed"
"...zee"
"This interview is over"
We need a phoneticphile video to sort this out
@@aceman0000099 "phoneticphile" That's a weird way to spell Tom Scott.
It's zed actually
@@vae3716 but more than 300 million people say it zee. So it's zee for US
I wonder, what do the rules say on whether or not that's a jinx?
"If you enlarge the domain of things you accept has a factorization then suddenly it becomes possible to factor." - Dr Eisenbud
To be fair, if you told a mathematician in the 16th century that x^2+y^2 factors into (x+iy)(x-iy) they would have told you
1. What are x,y, ^2 and i supposed to mean? We do math geometrically!
2. What square could have a negative area (regarding i)?
Generalising is what always improved math, and if you see something that doesn't generalise itself but is revolutionary, it relies on at least a few new generalisations to work, or it should have been realised way sooner
Yes, these are called field extensions.
I love modesty of mathematicians..they do not brag about their works, because they have no idea where to use it, they just love mathematics, that's all for them.
Yes. A mathematician knows he never knows everything.
@@duartesilva7907 he also knows that he can't know everything. It makes him sad, but that's the reality.
And, Bam! Jus like hat, he day after my 72nd birthday, I learned something new. Thanks Dr. Eisenbud, Numberphile, and UA-cam.
We see David again!
yay
Listening to him is so soothing. Also, I thought it’d be some familiar factorization from linear algebra, but it turned out to be much cooler!
If he ever wanted branch out, I can see him having a career in audio books with that buttery smooth delivery.
Sometimes I think this guy is too high level for this channel. But I wanna see more from him definitely
Fair enough, but I find Dr Peyam's channel even more challenging sometimes.
I guess it's better if you have some easy stuff and some hard stuff. Something for everyone.
I love Eisenbud's style! He has an ease of explanation that's very enjoyable to listen to.
I’m not saying it’s not easy or enjoyable, take my comment with a grain of salt, just the fact he can explain topics like this without losing the layman without dumbing down the mathematics and the fact that he is actually a contributor to pushing mathematics is awesome, and it shows not only in his enthusiasm but his work
I just mean that he is actually explaining topics on the cusp of his field, when a lot of these videos suffer from explaining things you would find in a typical course on various levels of mathematics, available on many other channels.... (not that that’s a bad thing either)... I meant it as a positive comment
Damn, I think it is just me missing numberphile's uploads frequently, but I was missing this guy. Such a nice person!
1:52 - "If you don't have enough tricks in your bag, put in a new trick" :-)
You kinda lost me halfway to the end, but I still watched it through, cuz it's interesting.
You have my admiration, I was lost the moment he started talking about matrices. XD
A gem per se (and especially in these troubled times). What a pleasure to watch Prof. Eisenbud. Thank you!
Such a great interaction with a very humble mathematician. It really is nice!
12:59 "Proving this depends on the theory of finite free resolutions, in which I'm an expert."
It feels like a bit of an understatement for Eisenbud to consider himself _only_ an expert on finite free resolutions :P
i read this comment as he said it
Why would it be an understatement? I don't know much about Eisenbud's work.
@@alazrabed Eisenbud literally wrote the book on commutative algebra.
@@alazrabed Sorry about the very late response! Eisenbud (and his collaborators, such as David Buchsbaum) proved some of the basic and foundational tools in studying finite free resolutions. He pretty much pioneered the topic!
I really enjoy hearing Dr. Eisenbud! :) Thanks for taking the time to make such wonderful videos.
I just got done with my Linear Algebra course, and you *had* to remind me of it just a few days later :P
Isn't it always nice to see that the stuff you learned is usefull? :)
Oh wow it's the TAs guy
bro u should watch linear algebra on 3b1b channel if you haven't
Same, I just finished a Bayesian Machine Learning course yesterday and thought I had seen my last matrix for a while!
No one is ever really finished with Linear Algebra :)
I understood nothing but I loved listening to him.
I was just procrastinating on a commutative algebra assignment and stumbled upon this video, not realizing this is the very David Eisenbud from the commutative book I was reading! (The book is great, of course.)
That was awesome. I really like David Eisenbud explanation, and that was an interesting conversation about his work.
Wow great! Def my favorite in linear algebra~ like the way you present it~
Early Numberphile videos talks about a specific number.
Nowadays Numberphile videos talks about partial derivatives and matrices.
.
.
.
.
Future Numberphile videos talks about hypertopology and combinatorial number theoy.
So very true!
I don't mind 😊
Honestly, I don't even have a clue what they are talking about
i hope
Numberphile in 2020s
This is mindboggling stuff. Kudos to Paul Dirac who only lived a mile or two down the road from where I am now!
He did the matrices portion very well. I enjoyed this alot and it makes me miss learning math. Thank you for this. He seems to be a very humble person.
Yesterday I was rewatching all of Professor Eisenbuds material on this channel and was hoping that there'll be more soon. Looks like my wish came true
I could listen to prof Eisenbud for hours. Thank you.
Every school in the World should have a David Eisenbud teaching math!
Enjoyed studying math and physics at Florida State University where Dirac spent his final years in semi-retirement (apparently he hated the humid summers compared to Cambridge but I bet the winters were much more enjoyable!). Many hours spent trying to understand analysis and algebra in the Dirac Science library.
Glad for you Stephen. Sounds like you took alot in in your course. You have a connection to one of the main men of the 20th century.
I Just love listening to Professor Eisenbud: he is crystal clear and surprisingly relaxing for me.
Love how naturals are represented by a hammer (you can't hit a nail a time and a half), rationals by an an axe (used to "divide" firewood), and complex numbers by a compass (referring to geometric interpretation).
What a brilliant educator. So humble and down to earth
I'd love to see more of Numberphile regulars explaining us part of their research.
This entire video was so heartwarming. I loved it.
Such a soothing voice and very interesting video as per usual.
True fact... saw thumbnail of David in my sub feed and was all like, "Aw hail, yeah!"... my favorite guest on Numberphile... and makes me wish I could have had him for a professor.
We all need more Eisenbud in our lives.
The inspiration from Dirac is really awesome. That guy was a genius. A random comment from him inspired Feynman's approach to quantum mechanics. And I use matrix factorizations at work all the time. This is wonderful.
It's not about mathématics only, Everybody listening here can appreciate , modesty, humbleness, altruism, soul beauty and a lot of hope for next scientist generations.
thanks for those precious minutes of pure pleasure.
"So citations are like your video views, then?"
More like "engagement statistics," since it only counts those people who have actually used your work to do further work.
Great video, Professor Eisenbud is great to watch. I would've liked to see more details though.
I'm introducing operator algebra (and factorization) to my Quantum Mechanics Students this week. I'm showing them this video because I find a nice introduction to the idea before we dive into some mathematics. Nice video.
Dr. Eisenbud makes this content so approachable
Started watching, watching took over, this Dr. got some chill charisma.
This is one of the best videos on this channel thus far
Another perspective on Dirac's equation is that it is factored using numbers from Clifford Algebra (a vast generalization of complex numbers, quaternions, and such).
Oh, I would have dearly loved to see a step-by-step worked example of this! Perhaps for a trivial-but-real case that illustrates the basic mechanism in a way that may fail to illustrate its depth, but still shows its utility.
Perhaps in a follow-up video?
This number domain expansion technique is especially useful during exams. Example: a kid gets an exam problem: divide 173 by 7. So the kid writes: "Let's extend the set of integers by a new number i, so that 7i=173. So the result of our problems is i". And this way he avoids the mentally exhausting process of actually solving the problem.
In addition to factoring matrices, you can meaningfully take their logarithms, exponentiate them and take a matrix to the power of another matrix.
Whaaaat? Really? How?
@@typo691 Taylor series. Those functions (exponential, log) can be represented as an infinite sum. And we now how to sum matrices.
I question taking a matrix to the power of another matrix. Sure, you can do A^B = exp(B ln A), but you could also do A^B = exp((ln A) B), as there's no guarantee that ln A and B commute. (There's also no guarantee that ln A exists - it doesn't, in general - but we can assume it does for the purposes of a definition.)
I must admit, the concept is new to me, and quite interesting. Thank you.
Thanks you. Really kinda clicked the relation between the SU(2) generating matrices and pauli's matrices.
I am really fond of Doctor Eisenbud's videos, and by proxy, of himself!
Mind, phew, blown. Yes, you’ve reached this audience, thanks for the enlightenment.
Great video and such a nice humble professor.
OMG that's Eisenbud?? The writer of one of my favorite books! ❤️❤️❤️❤️❤️❤️❤️❤️
Would you mind telling me what book it is?
I aspire to be at his level of chill.
Thanks Professor Eisenbud, I learned more about maths and physics history. You gave more than the maths ideas but also the fighting spirit to go farther :-)
Love the animations on this video
Even if I couldn't understand at first. He made me understand like magic. Great video from a nice guy. 😊
Could you make a video about Clifford algebra? It is a pretty cool way to simplify and unify a lot of mathematics in physics, and I think it deserves to be shown to larger audiences. Dirac's matrix problem in this video is basically Clifford algebra, but just with a matrix representation.
Nice one. I really enjoyed it.
this professor is a lovely teacher
Thank you David Eisenbud. You made it clear!
♥
This is unusually clear! Well, to my slow brain it is unusual to be able to follow along so easily. So, thank you.
As always, great video
Thank you for the video! All of you friends are super awesome! Oh moments with this video are sad.
Fantastic....always had this question in mind...nobody answered this way
This channel is 86% reason why I will quit my job and go for a PHD ... I can't live without this stuff ^^
I remember doing that little computation in particle physics. I didn't realise it was such an important mathematical concept.
Thanks for sharing this!
I'm currently working on rendezvous algorithm which uses quaternions to represent rotation of one object relative to other.But for initial "guess" there is affine approximation: we convert image of object into 2x2 matrix and 2x1 vector. And one of my tasks was to factor this 2x2 matrix into rotation, scale and "aspect" (looking from the side). So this video was very close to me: matrix factorization and also Dirac trick which has something to do with quaternions, though I still don't understand this connection thoroughly...
As soon as they mentioned Dirac in the context of the mathematical toolbox, I thought they might talk about the Dirac delta.
This is simply awesome!
“Dirac was satisfied. He invented matrix mechanics...” but I thought matrix mechanics was developed by Heisenberg.
He misspoke, I guess. What he shows in the video leads to the Dirac equation, a relativistic wave equation and not matrix mechanics. He is after all, as he says himself, not a physicist ;)
The whole motivation Dirac had was that the original relativistic wave equation, the Klein-Gordon equation, yields wave functions that cannot be transformed into probabilities. Taking the square root of it, so to say, would solve the issues but without considering matrix factorization there is just no way.
Matrix mechanics, from looking through Wikipedia, appears to be the early version if the Heisenberg picture. A refrence frame where you evolve operators instead of wave functions. With fixed wavefunctions, the formalism can be considered as working only with matrices (given a chosen basis).
Rififi50 yeah I was waiting for him to say Dirac introduced antimatter to interpret the solutions of the Dirac equation.
To be fair, he's not a physicist
I love Dr. Eisenbud❤️❤️
Thank you! Love this topic
He is just the guy I want to take classes om algebra ... He is heartwarming in his wise love to the area he is an expert of.
A Person With Exceptional Skill In A Particular Area❤❤❤.
I don’t get it , I almost watch every Numberphile Video on release , but this video didn’t show up in my feed. Might be the best video on yt I’ve seen in weeks. May the algorithm be with you for the next video . Love the Eisenbud Videos and hoping for another one with Clifford Stoll
Fascinating theorem!
Expetacular video
Just been into trouble with Unitary Matrix Decomposition for weeks and Now I see this in my recommendation......
Any use?
@@wierdalien1 No
Thank you Numberphile
7:57 I want him to add the parentheses so badly!!!!! This is torture!!!!
Why?
@@moodleblitz becaus xy-uv * A =/= (xy - uv) * A
I feel with you :)
@@worldOFfans But xy-(uv*A) doesn't really make any sense at all, so there's only one reasonable interpretation of xy-uv * A.
@@MuffinsAPlenty you're right, but you should not rely on reader doing the correctness work for you ;-) .
He has such a relaxing voice 😴
Yes, I want him to narrate an audiobook
I’m actually writing my essay on paraxial matrices in optics! Matrices are super convenient for simplifying complicated systems!
Great genius teacher thank you so much for your work
Could someone explain the connection between finding the root of xy-uv and finding roots of x^2+y^2+z^2+t^2? I don't see how it relates to complex numbers.
Let's say the first equation is rs - uv. We get the second equation if we set
r = x + iy
s = x - iy
u = z + it
v = -z + it
@@martinepstein9826 To get the equation with -t^2, set u = -z + t and v = z + t
I too felt like this was an important link that was missing.
Amazing! Never realized that a polynomial can be directly linked to matrix. Usually it is taught as a series of equations. It would be interesting to know any applications that prefer to turn matrix into ploynomials
He just talked about Dirac and how he applied it to quantum mechanics
It's also used in string theory
For square matrices, there's the characteristic polynomial, whose (ordinary numerical, i.e. complex) roots are the eigenvalues of the matrix. Interestingly, the matrix itself is a root of its characteristic polynomial.
Respect to Eisenbud, and his gigantic GTM Commutative Algebra
@@edawgroe It's a graduate-level text. At minimum you'd need to have had an undergrad abstract algebra course that tackled rings and fields.
Could we have more linear algebra on this channel please?
Great video!
It got me really curious: where can I find the algorithm to factor these polynomials?
Mind blows as we hear about String Theory!!!
Wow, cool video
i really like the theorem, also i like how he sounds like mr. burns
I loved "Nature just said, 'you should have been using matrices all along'"
I really love Professor Eisenbud videos. I would have loved to have him teach me mathematics (especially algebra). Is there any course from him online ? (PDF, Vidéos, etc.)
you can just buy his GTM, the thickest GTM of all
@@user-sq5hv9tj3i Lee's Smooth Manifolds is thicker IIRC
@@user-sq5hv9tj3i what's GTM?
@@Belioyt Graduate Texts in Mathematics. Prof. Eisenbud's "Commutative Algebra: with a View Toward Algebraic Geometry" is about 800 pages long.
Nice and useful
I love the hat and coat on the back of his door