Papa Fibonacci using Linear Algebra [ Recurrence Relations and Diagonalization ]

[@bjoernschermbach3957]

1) Papa Flammy, why do you keep misspelling Fiboinacci? ;) 2) 14:22 Eulergeddon best apocalypse! 3) I'd say 'so good' but I fear the BPRP revenge... 4) Can I call you Thomas Godchalk instead?

[@blazep5881]

Flammable Maths stick to papa, daddy just sounds... Weird

[@46pi26]

masterbaiter blaze dude my brother's name is Blaize (yes, with an i), and we would always call him the master baiter when we went fishing. What a coincidence!

[@dalitas]

You might not be the smartest boi but you are the flammiest

[@mishikookropiridze]

Papa fibonacci, linear algebra and cringe jokes ?! I am in heaven.

[@kgeorge7153]

seeing "linear algebra" in the title -> instant like

[@rohitg1529]

You're the only maths youtuber who goes at the perfect speed. Once your video is on 2x, I never need to pause to understand what you write, nor do I need to fast forward to see the next step.

[@youurdream182]

Nice to see the content of lectures I’ve went through becoming a meme 😂

[@youurdream182]

Flammable Maths, keep the spiciness up boi; love your vids papa ;*

[@robinros2595]

Nice! A small side-note: If you write T = S^(-1)AS, what you're actually doing is transforming your original space onto the column-vectors of S, performing A on that and then transforming back to the original space with base (1,0), (0,1). If you choose S to have eigenvectors of A for it's columns, you are actually performing the transformation of A on its own eigenvectors, so that's just multiplying the eigenvectors with their corresponding eigenvalues. So it's no surprise at all that T has the eigenvalues of A on its diagonal: this always happens (when A is diagonalisable). Hard to explain this over text, but when you see S and S^(-1) as transformation matrices it becomes totally clear that T should be a diagonal matrix and contains the eigenvalues of A :) Keep up the good memes :)

[@kgeorge7153]

Also it is not too hard to come up with the eigendecomposition one your own being exposed to the proper definition of a matrix-matrix multiplication: matrix on the left acting like a transformation on each column of the matrix on the right (then consider the matrix A times the matrix of its eigenvectors, and guess what's on the RHS). 3b1b did a great job explaining it, and it is probably the most crucial idea of the whole linear algebra, this makes everything reasonable, not artificial/coincidental.

[@kgeorge7153]

or may be not whole LA, but at least all decomposition problems -_-

[@damiandassen7763]

17:58 nice one "same spiel here"

[@ChrisLuigiTails]

0:45 - That's exactly what I said to my mother this morning! My parents discovered Fiboinacci last week and kept harassing me all week saying it's everywhere in nature

[@mathman6156]

I've just finished the first year of maths, and I'm loving the content you create, keep going!

[@maxwibert]

You can also check that phi conjugate is -1/phi, in which case the expression @23:36 simplifies to (phi^(n+1) - 1/phi^(n+1))/(phi^n - 1/phi^n). Because phi > 1, we have 1/phi^n approaches zero as n approaches infinity, so the fractional expression approaches phi^(n+1)/phi^n = phi.

[@omega_sine]

Another great video Papa Flammy. These videos are a great way to start off my morning.

[@sarthakchaudhary4375]

It's been 4 years and this is still the best video for this topic :)

[@deeptochatterjee532]

I saw your transformation matrix, and did the quick eigenvalue calculation in my head and instantly recognized the golden ratio equation. This is too good

[@noahholmes]

father of math, my golden papa

[@cameodamaneo]

I love the mini golden boi. I find him to be as useful as the popular golden boi a lot of the time.

[@46pi26]

Cameron Pearce I'm gonna start calling phi "Golden boi" from now on because of this comment

[@alexeipisacane7781]

Best video so far

[@Adam-lx3mt]

Alternatively, you can just seek a solution of the form F_n = a*phi^n for some phi. Thus phi must satisfy phi^(n+2)=phi^(n+1)+phi^n from the Fibonacci relation. Dividing through by phi^n and solving the quadratic gives us two solutions phi1 = (1+sqrt(5))/2 and phi2 = (1-sqrt(5))/2. Since both phi1 and phi2 satisfy the Fibonacci relation we can could express F_n as a linear combination of the two. So F_n = a*phi1^n + b*phi2^n. Finally, we note that phi1=-1/phi2 and use the boundary conditions F_0=0, F_1=1 to give a and b. This proof is quite a lot shorter and easier.

[@Dimiranger]

Everything strung wonderfully together, really good video!

[@francescoburgaletta3746]

Great! I'm finishing the exams for my first year and i just had the Linear Algebra one this february. To be honest i didn't like the subject and i still don't now, but it's somehow fascinating the perfectly coherent method used to solve many problems.

[@Arycke]

Awesome video. I love that the phi is independent of the initial integer conditions of 0 and 1 or 1 and 1 ( I guess so long as it's not 0 and 0) as most typically know from grade school. Great moves Papa Flammy, keep it up, proud of you :3

[@gamma_dablam]

Well ... it’s good that we don’t have Phi dependents here as we all know from Andrew’s channel how annoying those are

[@michaelempeigne3519]

This is great video about linear algebra and diagonalization. Until this day, I had no idea of how diagonalization was useful or how it worked.

[@adamcummings20]

I've started learning matrices in school so now I can finally understand this video. Epic

[@tajpa100]

Thank you for your wonderful lectures.

[@PapaFlammy69]

[@winterrain870]

Looks like Pell's Equations; And the matrix treatment was superb.

[@DiogoSantos-dw4ld]

Great video! I've done diagonalisation of matrices in my course but never seen why they're that useful but with the cancelling of the eigenvalues vector it makes so much sense! Usually though when constructing the S matrix I've been told to take the normalised eigenvectors, since you didn't do it I'm guessing it's just used to clean up S and S-transpose/inverse

[@noahgeller7018]

"phi's little brother" I love 3blue1brown too

[@i_deepeshmeena]

honestly a very informative video

[@MePatrick73]

Wow! I'm taking discrete math and we're learning how to solve linear homogeneous recurrence relations. We were only told to let an=r^n, find the characteristic polynomial, solve for the roots and then take a linear combinations to find the specifics values of the coefficients. This make so much sense now. Linear algebra in disguise xD !

[@TheNachoesuncapo]

this is going directy to my favs! great work men! really appreciate your work...

[@housamkak646]

That was an awesome video i wish u explain the linear algebra from zero

[@keroleswael9332]

Papa flammy destroying pure mathematics. Continue like this and prove more problems

[@holyshit922]

I like Papa Fibonacci with series But this approach is nice if we look for analogous to differential equation way There is also variation of parameters for difference equation You have Casoratian instead of Wronskian and summation instead of integration Homogeneous part can be written as system of equations and solved as you showed

[@mevnesldau8408]

Are you reading my minds? I LOVE linear algebra!

[@raph9054]

Papa is the best

[@jlue2051]

thank you gins i really enjoyed this one

[@46pi26]

Papa Flammy vs. 3b1b Who would win?

[@46pi26]

Flammable Maths He has transcended the realm of Papas and is now a Daddy Holy shit

[@ゾカリクゾ]

very interesting approach indeed

[@lucashunter6441]

Wow a Papa Flammy question that actually showed up on one of my hw assignments

[@Blacksun88marco]

0:42 TRIGGERED

[@jamieee472]

Wonderful Video!

[@emmanuelontiveros8446]

Both eigenvalues are the golden ratio

[@TheGarfield1337]

Im ersten Semester hat unser Prof. in der ersten LinA1- Vorlesung einfach nur aus Jux eine Herleitung für die Fibonnaci-Formel gezeigt, für die man keine Vorkenntnisse braucht. In der letzten Vorlesung hat er dann quasi den Kreis geschlossen und zum Schluss diese Herleitung gezeigt, um nochmal zu komprimieren, was wir gelernt haben. Fand ich mega cool damals als kleiner Erstsemester :D

[@cameodamaneo]

It's called the "Identity Matrix", my broski and we over here in New Zealand notate it as "I".

[@cameodamaneo]

Oh wow. I guess I'm not a very intentive boi.

[@owenmatwe2272]

What a legend.

[@andrijauhari8566]

Thanks papa flammy :)

[@ゾカリクゾ]

LOL dat euler at 14:30!! Definetely not expecting that

[@CreativeStyled]

This was so good.

[@sabhierules1]

I call it the Identity Matrix or a matrix of the standard vectors. 9:38.

[@fabiothezhao5518]

Damn Papa Flammy saved my ass on a discrete math problem that my instructor suggested (was abt finding a pseudo code (only integer operations allowed) for something with a structure similar to Fibonacci). Srs thanks!

[@nicholasleclerc1583]

I’m a pure math fan/geek, but I never really realized ‘till today how potent an useful Linear Algebra can be for just about freaking anything; and it made me feel like linear algebra was.... useful... Ywah, Ik, that doesn’t really sound pure math enthusiasm but more like Utilitarian blasé-ness, but I just never found *any* use better or on par ideas like calculus for some problems; I’ve even seen twice now, around a loooong interval of time and a long time ago too, a BlackPenRedPen video where a guest used linear algebra to solve a deeply complicated integral (calculus ITSELF was helped by this); a 2nd example of pure mathematics given a hand right here !

[@12346sandy1]

Love ur videos,keep it up!

[@twakilon]

I actually had to learn how to solve these type of problems for my Linear Algebrs course.

[@0707andy]

Oh yes, the O(logn) fibonacci is the best kind of fibonacci.

[@damianbuzon8119]

I love fibonacci .

[@xCorvus7x]

Beautiful.

[@fandeslyc]

Thanks ! Until now, i've never understood why there was a polynome

[@leif1075]

Why didbt You use FOIL Methid or quadratic formula at 8:50?

[@sujanbhakat1199]

Thank you

[@Applefarmery]

Lol i actually had this same exact problem in maths at the time this was uploaded

[@stenzenneznets]

Very nice

[@townsoncocke1670]

Forgive me for missing the reference in the video, but at 16:15 you replace two elements in the matrix with zeros saying those were our conditions for phi and phi conjugate. Could you refer me to when you explained those conditions for phi and phi conjugate that allowed that "substitution" (if that's the right word)? As this video is over a year old, this probably won't get a read, so I'll probably just end up calculating those elements to make sure they're zero. Anyway, great video! Thanks.

[@ortollj4591]

Hi Townson Cocke I did an other example with a 4x4 matrix , clik on the blue link on my comment

[@duncanw9901]

I have to admit my linear algebra computation ability is not exactly the best, I seem to have fallen into a strange gap between American schools teaching it in algebra 2/precalculus and not teaching it. Learned how to do it from your video though.

[@pappaflammyboi5799]

I'm a Flammy Boi fan.

[@midaskeijzer7107]

Next: integral of 1/(x+cos(x))? (or shouldn't give integral requests on non-integral video's?)

[@masteryoda1748]

C:90

[@user-pn9zm8qg7k]

talking to a camera for a long time must be quiet a work, a good demonstration of diagonalization.

[@WhiterockFTP]

@17:57 Same Spiel hier :D

[@phileasmahuzier6713]

So cool!

[@hassnataha9593]

If you want, can you explain Zeilberger's creative telescoping algorithm for definite hypergeometric sum, pleeeeeeeeeeeease

[@anon7692]

@ 2:07 "so it would be nice to work with some kind of matrix or vector" Is that ever nice?

[@anon7692]

i0.kym-cdn.com/entries/icons/original/000/023/021/e02e5ffb5f980cd8262cf7f0ae00a4a9_press-x-to-doubt-memes-memesuper-la-noire-doubt-meme_419-238.png

[@TheTim466]

1 - sqrt(5) I don' know if it's negative :thinking: ;)

[@HighInquisitorBonobotheGreat]

19:15 That look xD "ye boi ezy huh?"

[@koenth2359]

That was great and funny.

[@mohammedrahman9739]

Hi I am a math teacher in University of Garmian in Kalar a small part of the Kurdistan Region-Iraq. And I have a one problem [ Let a and b be two real number such that a

[@mohammedrahman9739]

@@PapaFlammy69 thanks its a good idea . thanks for your attention

[@azlanjor5019]

👏lit mafs

[@blazep5881]

you're trying hard to bring smoke Memes back. Cool

[@youngsandwich9967]

Could you do this with the natural log Fibonacci product like thing (each number in the sequence is the product of the preceding 2 numbers)

[@gammaknife167]

I'm sure many other people will tell you this too - it's the identity matrix. Loving this week, especially this one!

[@GreenMeansGOF]

It’s usually denoted by capital i with a subcript of n where n is the size of the matrix(2x2, 3x3,...). P.S. I hate it that capital i looks like lowercase L.

[@deeptochatterjee532]

Isn't the diagonalized matrix in general just the eigenvalues on the diagonal?

[@almightyhydra]

Well, you assumed the fact that S is (v1 v2), so I think it's also fine to assume T is [e1 0; 0 e2]. They go together really. If you wanted to prove that you should have done the whole proof of the P^-1AP = D diagonalization process. ^_^

[@maxblack493]

Salute.

[@shawon265]

I appreciate what you did, but all the calculation is a little bit frustrating to me. That's why I never like Linear Algebra in my Engineering undergrad life. But thanks to 3b1b’s visualization techniques, I could tell you the diagonalized matrix right after you figured out the eigenvalues. Basically T is describing the same A matrix but in eigenbasis. So, the diagonal elements must be the eigen values indicating the scaling factor.

[@thegrb93]

Can the same be done with the Mandelbrot set's recurrence relation?

[@cameodamaneo]

2:23 Please commit to your jokes in the future thanks.

[@The2bdkid]

In diagonalization, the T is by definition the eigenvalues on the diagonal, right? That's how I was taught at least.

[@GhostyOcean]

Hehe, "tongue twister" became "tongue breaker"

[@GhostyOcean]

Flammable Maths tongue breaker makes more sense to me, also more fun to say

[@ChrisLuigiTails]

That's because in Germany they say "Große Kartoffel" and it litterally translates to "tongue breaker" and I think that's beautiful

[@janlange6416]

@@ChrisLuigiTails omfg lololol

[@SugarBeetMC]

13:10 Dat S.

[@thomasblackwell9507]

I thought that this was going to be about linear algebra. Herr Professor Papa Flammy all I can say is “NICHT SHIZEN!” You say you are stupid, I would hate to think of an evil smart you!

[@WoWSchockadin]

So as Fibonacci is short for Filius Bonacci, which means son of Bonacci will the son cancel out with the papa and leave: Papa Fibonacci = Bonacci? :-D

[@MuitaMerdaAoVivo]

Papa, you're the best! Love your channel!

[@ThePron8]

just one question: why didn''t you simply wrote T when you evaluated the eigvalues? I mean, the diagonal matrix expressed in terms of the eigvectors basis is (for construction) the matrix with the eigvalues on the diag, then maybe you wasted a lot of work 😂

[@thechannelofeandmx4784]

Papa flammy, can you integrate mah boi here? sin(2pi*sqrt(1-x^2))

[@peterdriscoll4070]

Yeah. Complicated way of getting this result. But cool. phi to the power of n is a solution of the recurrence relation.

[@almightyhydra]

7:18 "chi" is pronounced "kai" (hard K sound and rhymes with "pie"). Also, the identity matrix is more commonly denoted as I rather than the "blackboard" 1.

[@bamdadshamaei1415]

What does pre record mean?

[@matteodamiano6733]

Papa Lucas when

[@46pi26]

Anyone who loves Papa Fibonacci needs to check out the song Lateralus. Also Papa Flammy>Papa Fibonacci

[@46pi26]

No songs for Papa Flammy tho :/

[@cameodamaneo]

I hate the shoehorned mathematics in that song. The lyrics are pretty good though.

[@46pi26]

Cameron Pearce Yeah Maynard himself said he regretted it but I personally really like the riffs. Not because I'm a sheep and believe they somehow spiritually resonate with me, just because it sounds pretty badass.

[@46pi26]

Flammable Maths I think you forgot to apply the negative signs at some point after the Papa operator. It's actually Papa Flammy>Papa 46&pi

[@elfaroukharb3979]

Papaaaa

[@thedoublehelix5661]

Wow

[@gnikola2013]

0:34 Papa? Is that you? PAPA Y U NO PAY ATTENSHONE TO ME