Matrix Factorization - Numberphile

[@ianprado1488]

Dr. Eisenbud seems like such a nice guy

[@numberphile]

He *is* a nice guy.

[@andrewzhang8512]

@@numberphile Why do you seem serious?

[@MuffinsAPlenty]

He absolutely is!

[@vgstep]

Totally

[@shmunkyman33]

"DR. EISENBUD IS INDEED A HUMAN WHO IS NICE" *blinks "HELP" in morse code*

[@adamweishaupt3733]

According to Google Scholar, "Homological algebra on a complete intersection, with an application to group representations" has 678 citations.

[@andrewzhang8512]

huh

[@PHDnHorribleness]

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.

[@CommodoreHorrible]

@@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"

[@samuelthecamel]

@@CommodoreHorrible You should write a paper on it and then do the calculations on your own paper.

[@martinpaddle]

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).

[@penisbreath952]

love his answer at 15:01 "It makes me pleased, that's all really." :)

[@randynguyen9006]

@@ryanhenrydean1584 Thanks for pointing out his Username

[@davidr2421]

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.

[@bonob0123]

no i think it went the other way. He wrote the paper first and then the physicists found it useful and it became popular.

[@Isiloron]

@@bonob0123 David Vaughan was talking about the generalization paper, not the initial paper.

[@marcoantonio7648]

i know right?

[@bonob0123]

@@Isiloron Fair enough

[@NYsummertimeCHI]

@@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?"

[@crimsonkhan3815]

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.

[@duartesilva7907]

Yes. A mathematician knows he never knows everything.

[@stv3qbhxjnmmqbw835]

@@duartesilva7907 he also knows that he can't know everything. It makes him sad, but that's the reality.

[@sb_dunk]

12:23 "So the reason that x was ok here is because it was multiplied by..." "...zed" "...zee" "This interview is over"

[@aceman0000099]

We need a phoneticphile video to sort this out

[@antagonistictherapy]

@@aceman0000099 "phoneticphile" That's a weird way to spell Tom Scott.

[@vae3716]

It's zed actually

[@stv3qbhxjnmmqbw835]

@@vae3716 but more than 300 million people say it zee. So it's zee for US

[@PeterBarnes2]

I wonder, what do the rules say on whether or not that's a jinx?

[@ebrahimsonday5941]

"If you enlarge the domain of things you accept has a factorization then suddenly it becomes possible to factor." - Dr Eisenbud

[@JonathanMandrake]

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

[@KilgoreTroutAsf]

Yes, these are called field extensions.

[@inyobill]

And, Bam! Jus like hat, he day after my 72nd birthday, I learned something new. Thanks Dr. Eisenbud, Numberphile, and KZbin.

[@yorickdewid]

We see David again!

[@andrewzhang8512]

yay

[@khalidbinwalid1566]

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!

[@RaunySilva]

Damn, I think it is just me missing numberphile's uploads frequently, but I was missing this guy. Such a nice person!

[@neonblack211]

Sometimes I think this guy is too high level for this channel. But I wanna see more from him definitely

[@rogerkearns8094]

Fair enough, but I find Dr Peyam's channel even more challenging sometimes.

[@GruntDestroyarChannel]

I guess it's better if you have some easy stuff and some hard stuff. Something for everyone.

[@mobius32]

I love Eisenbud's style! He has an ease of explanation that's very enjoyable to listen to.

[@neonblack211]

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

[@neonblack211]

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

[@neon-rust]

If he ever wanted branch out, I can see him having a career in audio books with that buttery smooth delivery.

[@prikarsartam]

Such a great interaction with a very humble mathematician. It really is nice!

[@romanbykov5922]

You kinda lost me halfway to the end, but I still watched it through, cuz it's interesting.

[@TakeWalker]

You have my admiration, I was lost the moment he started talking about matrices. XD

[@danielurbinatoro9496]

A gem per se (and especially in these troubled times). What a pleasure to watch Prof. Eisenbud. Thank you!

[@kwgm8578]

David, we engineers may not be writing many papers, but we have used and appreciated your brainchild. Thank you.🧙🏼‍♂️💙

[@Carbon-XII]

1:52 - "If you don't have enough tricks in your bag, put in a new trick" :-)

[@shaileshrana7165]

I understood nothing but I loved listening to him.

[@NeoLogicification]

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.

[@martinepstein9826]

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

[@Miyelsh]

@@martinepstein9826 To get the equation with -t^2, set u = -z + t and v = z + t

[@cretinobambino]

I too felt like this was an important link that was missing.

[@MuffinsAPlenty]

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

[@Lastrevio]

i read this comment as he said it

[@alazrabed]

Why would it be an understatement? I don't know much about Eisenbud's work.

[@selenamertvykh6481]

@@alazrabed Eisenbud literally wrote the book on commutative algebra.

[@MuffinsAPlenty]

@@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!

[@ubertoaster99]

This is mindboggling stuff. Kudos to Paul Dirac who only lived a mile or two down the road from where I am now!

[@alaanasr7505]

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.

[@narutosaga12]

So very true!

[@aleksapetrovic7088]

I don't mind 😊

[@recoveryemail1046]

Honestly, I don't even have a clue what they are talking about

[@yigitsezer6696]

i hope

[@jamirimaj6880]

Numberphile in 2020s

[@Axacqk]

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).

[@nataliaquinones4356]

I really enjoy hearing Dr. Eisenbud! :) Thanks for taking the time to make such wonderful videos.

[@drpkmath12345]

Wow great! Def my favorite in linear algebra~ like the way you present it~

[@mathhack8647]

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.

[@davidgillies620]

In addition to factoring matrices, you can meaningfully take their logarithms, exponentiate them and take a matrix to the power of another matrix.

[@typo691]

Whaaaat? Really? How?

[@lumer2b]

@@typo691 Taylor series. Those functions (exponential, log) can be represented as an infinite sum. And we now how to sum matrices.

[@tomkerruish2982]

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.

[@Cobalt0000000]

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.)

[@charlieangkor8649]

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.

[@stephenbeck7222]

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.

[@denisdaly1708]

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.

[@DyllonStejGaming]

I just got done with my Linear Algebra course, and you *had* to remind me of it just a few days later :P

[@yrrgallerte354]

Isn't it always nice to see that the stuff you learned is usefull? :)

[@FtwXXgigady]

Oh wow it's the TAs guy

[@victorarturoibanezaliaga8783]

bro u should watch linear algebra on 3b1b channel if you haven't

[@jledragon]

Same, I just finished a Bayesian Machine Learning course yesterday and thought I had seen my last matrix for a while!

[@brightsideofmaths]

No one is ever really finished with Linear Algebra :)

[@ImKurono]

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.

[@WaffleAbuser]

7:57 I want him to add the parentheses so badly!!!!! This is torture!!!!

[@moodleblitz]

Why?

[@worldOFfans]

@@moodleblitz becaus xy-uv * A =/= (xy - uv) * A

[@brightsideofmaths]

I feel with you :)

[@MuffinsAPlenty]

@@worldOFfans But xy-(uv*A) doesn't really make any sense at all, so there's only one reasonable interpretation of xy-uv * A.

[@ericbischoff9444]

@@MuffinsAPlenty you're right, but you should not rely on reader doing the correctness work for you ;-) .

[@exxzxxe]

Every school in the World should have a David Eisenbud teaching math!

[@nightworg]

That was awesome. I really like David Eisenbud explanation, and that was an interesting conversation about his work.

[@michelebotticelli3258]

I Just love listening to Professor Eisenbud: he is crystal clear and surprisingly relaxing for me.

[@roderickwhitehead]

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.

[@sarahcogswell9166]

Dr. Eisenbud makes this content so approachable

[@andrewxc1335]

"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.

[@paulpantea9521]

We all need more Eisenbud in our lives.

[@CosmiaNebula]

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).

[@lambda494]

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.

[@DerNesor]

This channel is 86% reason why I will quit my job and go for a PHD ... I can't live without this stuff ^^

[@1978Maedhros]

OMG that's Eisenbud?? The writer of one of my favorite books! ❤️❤️❤️❤️❤️❤️❤️❤️

[@johnchesterfield9726]

Would you mind telling me what book it is?

[@flymypg]

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?

[@thatdude_93]

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

[@davidianmusic4869]

Mind, phew, blown. Yes, you’ve reached this audience, thanks for the enlightenment.

[@javierantoniosilva8477]

I'd love to see more of Numberphile regulars explaining us part of their research.

[@manueldelrio7147]

I am really fond of Doctor Eisenbud's videos, and by proxy, of himself!

[@tombulous]

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.

[@TheTwick]

I could listen to prof Eisenbud for hours. Thank you.

[@kaakatin]

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.

[@daniellanes813]

Started watching, watching took over, this Dr. got some chill charisma.

[@bittertea]

I aspire to be at his level of chill.

[@redambersoul]

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.

[@matiasreinoso3393]

This is one of the best videos on this channel thus far

[@thederivationchannel4243]

What a brilliant educator. So humble and down to earth

[@kdpwil]

At 11:59 in the video: Couldn't you write x+y^2 as x*z+y^2, factor that using matrices, and then set z=1 to obtain a factorization of x+y^2?

[@王珂-k7d]

Respect to Eisenbud, and his gigantic GTM Commutative Algebra

[@selenamertvykh6481]

@@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.

[@allmycircuits8850]

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...

[@Celastrous]

At 8:48, how does multiplying two 2x2 matrices line up with a 4x4 matrix? The matrix squared should stay 2x2 and diagonal, with squared elements only.

[@MuffinsAPlenty]

They aren't 2x2 matrices he's multiplying together. Those are "block matrices". Remember that A and B are both 2x2 matrices.

[@ericpowell96]

He has such a relaxing voice 😴

[@Belioyt]

Yes, I want him to narrate an audiobook

[@Ridley_Prime]

Couldn’t you just substitute a new variable x = y^2 any time you have a linear term? Could you apply this to things of a non-positive integer order, like x^y, or x^4.87 or x^-2?

[@skylardeslypere9909]

I've got a question about his theorem. If you do allow the matrix factorization to include constants, does it mean we CAN factor any and every polynomial? Take P(X,Y) = X+Y² for example. If we 'treat' it as another polynomial P(X,Y,Z)=XZ+Y², factor that one without constants, and plug in Z=1, do we not get a factorization?

[@chunchen3450]

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

[@Belioyt]

He just talked about Dirac and how he applied it to quantum mechanics

[@Belioyt]

It's also used in string theory

[@tomkerruish2982]

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.

[@senkottuvelan]

Even if I couldn't understand at first. He made me understand like magic. Great video from a nice guy. 😊

[@danibarack552]

I would have liked to see him actually factorize the polynomial hr started with..

[@mananself]

“Dirac was satisfied. He invented matrix mechanics...” but I thought matrix mechanics was developed by Heisenberg.

[@Rififi50]

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).

[@mananself]

Rififi50 yeah I was waiting for him to say Dirac introduced antimatter to interpret the solutions of the Dirac equation.

[@MatthijsvanDuin]

To be fair, he's not a physicist

[@digitig]

I loved "Nature just said, 'you should have been using matrices all along'"

[@gabrielhermesson9926]

As soon as they mentioned Dirac in the context of the mathematical toolbox, I thought they might talk about the Dirac delta.

[@MrYashraj]

A Person With Exceptional Skill In A Particular Area❤❤❤.

[@kiro9291]

this professor is a lovely teacher

[@trevorteolis3691]

Great video, Professor Eisenbud is great to watch. I would've liked to see more details though.

[@foxtrot.tango.whisky]

Can we get a video on probing variation of the fine-structure constant using the strong gravitational lensing? Please. Thank you.

[@SimpMaker]

Thanks you. Really kinda clicked the relation between the SU(2) generating matrices and pauli's matrices.

[@T75-n1m]

Just been into trouble with Unitary Matrix Decomposition for weeks and Now I see this in my recommendation......

[@wierdalien1]

Any use?

[@T75-n1m]

@@wierdalien1 No

[@meexi9824]

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

[@bittertea]

Could we have more linear algebra on this channel please?

[@thescienceprint5825]

David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and was Director of the Mathematical Sciences Research Institute from 1997 to 2007. He was reappointed to this office in 2013, and his term has been extended until July 31, 2022.

[@sbmathsyt5306]

Such a soothing voice and very interesting video as per usual.

[@omarsamraxyz]

I love Dr. Eisenbud❤️❤️

[@mehfak]

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.)

[@王珂-k7d]

you can just buy his GTM, the thickest GTM of all

[@alicewyan]

@@王珂-k7d Lee's Smooth Manifolds is thicker IIRC

[@Belioyt]

@@王珂-k7d what's GTM?

[@MuffinsAPlenty]

@@Belioyt Graduate Texts in Mathematics. Prof. Eisenbud's "Commutative Algebra: with a View Toward Algebraic Geometry" is about 800 pages long.

[@thescienceprint5825]

Paul Adrien Maurice Dirac was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics.

[@cwaddle]

So what is the application of the matrix factorization? The traditional polynomial factorization will tell you where the zeros are, but does the matrix factorization do the same thing?

[@dominiquelaurain6427]

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 :-)

[@kcmichaelm]

This entire video was so heartwarming. I loved it.

[@abhinavagarwal244]

Fantastic....always had this question in mind...nobody answered this way

[@braedenlarson9122]

I’m actually writing my essay on paraxial matrices in optics! Matrices are super convenient for simplifying complicated systems!

[@mufasao6776]

"If you want to solve an equation like 3x-1=0, you can't solve that in integers, so you invent rational numbers (fractions), and then you suddenly can solve it; x is 1/3. Or, if you said 3x+1=0, then you'd have to know about negative numbers too. And for a while, negative numbers were sort of very strange things in mathematics. Then they got ordinary, and we're happy to use them." "Nature somehow follows along, or, really, nature was ahead of us there, I think. So nature knew about complex numbers, but didn't bother to tell us for a long time. And then we needed them for something, and we realized that they were useful, and now are the basis of lots of physics and everything. So they're really out there in nature, even though they're called imaginary or complex."

[@MCLooyverse]

Does this mean that you could take any 5th degree polynomial (in x), multiply it by x^2 (such that it has no 0th or 1st degree terms), then factor that? I suspect that I don't fully understand the matrix representation, or don't fully understand the process, but it seems that you could effectively use this to find the roots of a 5th (or higher) degree polynomial.

[@digitig]

That's my understanding of what he said, although He said the process was algorithmic, which isn't the same as there being a formula. Also I wonder whether there might be an issue getting rid of that unit matrix ("Curses! I could have got away with it if it hasn't been for you pesky unit matrix!")

[@zozzy4630]

I like that now we know how to find the square root of a matrix, but he still never explained how xy-uv is actually the same as -c^2(d^2/[dt^2]) + (d^2/[dx^2]) + (d^2/[dy^2]) + (d^2/[dz^2]). I'm sure it's something he's taking for granted that linear algebra students already know, but now I'm just confused and lost. Can we get a video about that sometime?

[@ycu4AB]

i don't know the analogy of the differential operator to the polynomial t^2+x^2+y^2+z^2. but this polynomial can be factored into matrices, because it is degree 2 (as he later explains). the xy-uv stuff is only an interlude to motivate matrix factorization. there is no relation to the t^2+x^2+y^2+z^2 polynomial.

[@nicolasboyardi9382]

Maybe it has something to do with changing the basis, but is just a quick thought I got.

[@zozzy4630]

@@ycu4AB Oh, thank you! That was the main thing that had me stuck; I didn't realize he was using it as an analogy. I suppose xy-uv is probably the simplest multi-term polynomial of all terms with degree greater than 1, and proving that that works is enough to prove that the 4D quantum equation works, too. I'm still a bit lost on that first step as well, but I think with enough time and pencil and paper I could probably figure it out - my intuition tells me you probably have to integrate it and then re-derivate it a few times and eventually a factor of 1 pops out of one of the terms or something.

[@vs-cw1wc]

It's called lightcone coordinates and has applications in string theory. look that up.

[@RalphDratman]

This is unusually clear! Well, to my slow brain it is unusual to be able to follow along so easily. So, thank you.

[@jpalreis]

Great video! It got me really curious: where can I find the algorithm to factor these polynomials?

[@legendhero-eu1lc]

Thank you for the video! All of you friends are super awesome! Oh moments with this video are sad.

[@titan-ishan1358]

301 views stuck was a legendry video🤗🤗🤗🤗🤗🤗LOVE FROM INDIA

[@evenprime1658]

okay i see my mans coming up with more quadratic formulas

[@lamgam-ts8tr]

Dr. Eisenbud is a treasure

[@jhhtaylor]

Love the animations on this video

[@Ricocossa1]

I remember doing that little computation in particle physics. I didn't realise it was such an important mathematical concept.