A Number to the Power of a Matrix - Numberphile

[@TomRocksMaths]

Coming soon - a number to the power of an operator... oh hello Quantum Mechanics.

[@darealpoopster]

just exponentiate hamiltonians and be happy, physicists are insane

[@LordMarcus]

Every good programmer knows to keep data and code separated. 😆

[@sirusdesnoes3094]

It’s so cool to see a math topic I’m learning in class line up with a new Numberphile video, can’t wait to see eigenvectors and eigenvalues!

[@jacquesgastebois]

@@sirusdesnoes3094totally agree with you

[@aminzahedim.7548]

As someone who has passed QMI,II&III, Real Analysis I&II, and Complex Analysis (to some extent), what I’d love to see is a “rigorous” definition of e to the power of a matrix, not just using the taylor expansion of e^x; and then some: the criteria for a matrix to qualify as a power, how a finite-dimensional matrix is upgraded to a full-fledged infinite-dimensional operator (such as the Hamiltonian), the intuition behind and proof of relations like det(e^A)=e^tr(A), the exponential relations in cases where the commutator [X,Y] doesn’t vanish, how eigenfunctions and eigenvalues work (especially in the case of Hermitian operators), how functional analysis is rigorously done, and how it all relates to specific subfields of manifold theory and topology. I realize it’s not quite possible to cover all this in a single video, but I just love the subject matter as well as your way of presenting it so much 🙏🏻

[@Galva94a]

I think Tom missed a great opportunity for talking about how COMPLEX NUMBERS could be represented with matrices (where the imaginary unit "i" is the matrix (0,-1,1,0) The formula then becomes euler's formula: e^iwt=cos(wt) + i sin(wt)

[@qwertek8413]

nice

[@Datamining101]

coming soon to a numberphile near you

[@skergy1579]

The relation of complex numbers here also makes sense considering the resulting matrix of e^A is also just the 2-dimensional rotation matrix!

[@15silverblade]

Very well pointed

[@asherdp]

i noticed where it was going as soon as he said "(0, -w, w, 0)"

[@ohadcohen9813]

I was missing the explanation that this is the 'rotation operator'. This links to many results in physics where either a vector is rotated about the origin, or a mass oscilates with a fixed frequency 'w'.

[@DeclanMBrennan]

And shows up in Electrical Engineering (Alternating Current etc).

[@ebenolivier2762]

And also in 3D computer graphics

[@hj45lp]

Exactly! 👍😎

[@javen9693]

and this is because the matrix A he chose is secretly a matrix version of the imaginary number i. so e^Ax is similar to e^ix, but it keeps the complex number stuff hidden

[@tetsi0815]

Was about to write something similar. The e^(something) links soooo many seemingly different things in mathematics, engineering and computer science to the point where one is surprised if e^(somehing) does not show up somewhere.

[@dikkedorus]

The thing that I was missing is that the matrix you get is actually the 2D rotation matrix of omega degrees. As a graphics programmer I recognized it pretty early on and I thought that that was where it was going. Effectively you can multiply a vector with that matrix to rotate it along Origin; this is used extensively in 3D computer graphics and animation. It was probably the first matrix I learned doodling away as a teenager.

[@jacktremblay71]

Was going to mention it too!

[@HyperCubist]

omega radians ;)

[@ffggddss]

@@HyperCubist Actually, radians/second, with t as time in seconds. Or with t in any time units, and ωt in radians. Fred

[@WormHunter]

So that's why it seemed so familiar.

[@hellsboy96]

The explaination of where t comes from was kind of weird, as we could have replaced w*t with any variable from the beginning, however it becomes more clear if you apply that matrix to a vector (e^At * v ). For this specific Matrix that would be a rotation in 2D space around the origin by an angle w*t! w and t are often used variables in physics, where w is angular velocity and t time, so w * t represents the angle after spinning something at a specific velocity a given amount of time.

[@albericponcedeleon2696]

I was feeling a bit lost with what the operation actually was (not a math major). The idea of w*t representing a rotation makes it more concrete in my mind.

[@ErikScott128]

This is the beginning of linear multi-variable differential equations. In practice, you never actually have to do any of this math again once you've proven it, but it's still important. Steve Brunton here on KZbin has some excellent lectures on this stuff, and it's honestly a much better introduction to differential equations than what I had in undergrad.

[@JesseHersch]

I remember this from quantum mechanics! Seemed crazy at the time but you get used to it. Like that von Neumann quote: "in mathematics you don't understand things. You just get used to them" I think he understood them though.

[@radiac6818]

Rumor has it that famed theoretical physicist Paul A. M. Dirac first figured this out, in his head, while attending a cocktail party.

[@hoebare]

I'll have what he (Dirac) is having.

[@StormwaterIsOneWord]

Even after 13 years of watching, the videos still amaze me. Fantastic work and effort as always, thanks for sharing with us!

[@pixelguitarman]

As an Electrical Engineer, I feel right at home in this video 🤣

[@oakpope]

Was about to write something along the same line. Physics for the win !

[@CheckmateSurvivor]

As an electrical engineer I am also the inventor of Imaginary (Complex) Chess, the most complex chess variant ever invented.

[@user-yt198]

EE151 Circuit Theory ☺

[@MAgy9ko]

I can feel your pain bro.

[@allasar]

"A number to the power of a matrix" sounds like "coloring a moral compass with ice cream flavors" to me.

[@jespervalgreen6461]

Indeed, and if you ever find a way to make operational sense of that proposition, I'm all in!

[@rnts08]

​@@jespervalgreen6461graphics programming :)

[@Marinealver]

Well can you do a reman sum where n =/= an integer (i.e. 2.5).

[@benthayermath]

And the craziest part is we actually use this very regularly! It came up on my graduate quantum mechanics final :)

[@agrajyadav2951]

cry about it

[@FScott-m1n]

The flow and clarity of Tom's lectures have improved immensely.

[@ernstfrutphlinguhr2494]

He still suffers from employing enthusiasm in lieu of clarity, or more precisely his enthusiasm makes him jump or gloss over details that would really benefit from further explanation.

[@youneskafie2008]

OMG this was crazy, really nice when you can go from linear algebra to calculus without expecting it.

[@msclrhd]

The cos + sin form is also euler's expansion of e in matrix form with (0 -1|1 0) being i! And IIRC the derived matrix for e is a rotation matrix, which also makes sense as euler's cos+sin expression is essentially defining a rotation at a given angle! It's nice when different areas of mathematics link up with each other.

[@beginneratstuff]

Wow... it's very satisfying to be at a point in my math education where I can watch this and understand everything. I just took my linear algebra final exam today. It's cool that the final matrix is a rotation matrix. Reminds me of how Euler's formula relates to rotations in the complex plane! The matrix differential equation was cool too. I hadn't seen something like that before.

[@Klaevin]

this only confirms my belief that mathematicians are just 5 year olds bashing together their toys and coming up with the result in their heads

Yes, that's largely what real maths is. Which makes it so ironic and sad that it's taught in such a boring, rigid, and off-putting way in schools. If schools taught what mathematicians actually do, I'm sure many more people would be into it.

[@eltiess]

Shhh you'll have people smashing all their favourite toys together soon

[@MattMcIrvin]

This stuff in this video turns out to be super important in physics. They play these games and useful tools come out, somehow.

[@timmy18135]

Not as difficult as the math in chaos theory or nullo space !

[@harlanweid7723]

Interesting fact, if you treat w as rotation angle, exp[A] will give you matrix of rotation by that angle. It works in 3D too with rotation vector and corresponding antisymmetric matrix.

[@peppermann]

Class 😎👍🏻

[@_Babak]

I love the fact that you math guys get so excited about the abstract beauty of it, while you do not care about all the electrical enginieering you ignored here :P

[@davidgillies620]

This is also related to Pauli matrices which turn up in things like spinors and Lie algebras. The exponential form e^A is the 2D rotation matrix, which corresponds to the group SO(2), very important in the study of light polarisation, and spin in quantum mechanics.

[@Entropic_Alloy]

Matrix Exponential was something that was drilled into us during grad school engineering.

[@LonkinPork]

instead of grad school I went straight into the field, so I missed out on a lot of the cooler math concepts 😭

[@steffenbendel6031]

@@LonkinPork quantum field theory or normal field theory?

[@Icytroll]

Yea, fond memories of the 2D rotation matrix from a course on computational dynamics, learning that it simply represents rotating your x- and y basis vectors around the origin of a coordinate system, and how useful of a tool that is when solving any sort of computational motion problem

[@seedmole]

As someone who got into mathematics on a personal level through music and then game development, this is great. Rotation matrices, producing sine and cosine functions? Right on.

[@davidiverson5928]

"Why the tea?" British people always need tea.

[@Micetticat]

The final calculus "trick" was just amazing

[@AgentM124]

What's next, a matrix root? Matrix factorial?

[@landsgevaer]

Once you have diagonalized the matrix, it is all easy.

[@AgentM124]

@@landsgevaer thatis, if your matrix is diagonalizable

[@BlazeDragonX]

You can always use continuity though! The diagonalisable matrices are dense in the set of all matrices!

[@gustavk3227]

@@AgentM124matrix square root can be defined more generally with jordan decomposition and the cauchy integral formula

[@haraldmilz8533]

The factorial of any square matrix with real or complex coefficients is in fact the null matrix. See also "nilpotent".

[@gowzahr]

You know he's a real mathematician because when asked about the 't', he didn't talk about any of the real world applications, he instead took the opportunity to talk about more math.

[@quinnencrawford9707]

Its gonna equal pi, isn't it.

[@quinnencrawford9707]

I mean, a matrix can't equal pi since it's a scalar, but, it's gonna be something related, calling it.

[@canyoupoop]

πI you mean?

[@JacobPlat]

​@@quinnencrawford9707no.

[@abigailcooling6604]

@@quinnencrawford9707 Weellll... since this matrix multiplication is related to rotations, which are related to circles, which are related to pi, I guess you're kinda right?

[@timmy18135]

Or wau

[@riuphane]

Still struggling to understand how or why this would be a valuable operation, but trying to think about it is extremely interesting

[@chrisglosser7318]

Basically this is how the math works for the Heisenberg picture of quantum mechanics (extremely useful)

[@javen9693]

it is the same as e^ix which is how computers rotate things, like in 3d graphics. when you watch Toy Story 7 and see Woody *rotate* his arms to the voice-acting of an AI copy of Tom Hanks, you'll have e^At to thank for it.

[@HyperCubist]

@@javen9693 Although Woody probably uses quaternions.

[@Marinealver]

Computer Science and Quantum Physics. And if Crypto-currencies gain liquidity and become legal tender, the financial sector as well.

[@ErikLeppen]

By the way, what's interesting, now you have a definition for e to the power of a matrix, you can also take 91 to the power of a matrix by defining 91^A as e^(A * ln(91)). Where, of course, 91 is not special, so you can now define any number to the power of a matrix.

[@the_eternal_student]

The best numberphile video I have seen in a long time.

[@KIPTOOAMMONBELYON]

I should go back to Mathematics. This is fun.

[@IsomerSoma]

If you want to know more look up 'functional calculus' and the spectral theorem of which this e^A is an example.

[@mobgamerz7397]

I love to play with numbers in mathematics. There is another level of enjoyment by solving these types of problems.

[@Jaylooker]

At 13:03 the result is just the Lie group SO(2) of special orthogonal matrices satisfying a special determinant det = 1. Lie originally derived his theory of (continuous) Lie groups as solutions to differential equations.

[@nullgeodetic]

You can make it look mystical by saying you are "raising a number to the power of a matrix" or you can say that you are calculating the exponential of a matrix (if well defined). The number e is the exponential applied to 1.

[@robertolson7304]

When you square a square in regular reality. You are just doing. - 1 x -1 =1 and 1x1=1. You will not find 1 x -1 = -1 or -1 x 1 = -1 because they are not parallelograms. 3,6,12, 24 binary. 3, -6 , 12, -24 are odds.. that 1 odd number keeps coming up.

[@polbecca]

I'm glad I spent five years doing mechanical engineering just so I can understand a Numberphile video 25 years later. 😂

[@Kris-P_Boi]

What an incredible video! Really shows the deep connections that exist between different branches of mathematics.

[@chemistrycuber]

At 13:13 this matrix represents a rotation anti-clockwise of wt degrees. Very nice video as it encompasses lots of functions, expansions, matrices and calculus!!

[@amirilan4435]

I remember encountering it for the first time in quantum mechanics course, i was so confused haha

[@Wolf-if1bt]

It seems A is a generator for rotations in the plane. Could you explain the relationship with Lie algebra and Lie group?

[@falkranduhm10]

I watched this video twice. Must be glitch in the matrix. It certainly was not because I did not get everything the first time.

[@shield543]

At 12:18 the matrix on the right is the Pauli Y matrix multiplied by i (imaginary number) 😄 All of the maths in the video is very important for quantum mechanics, quantum computing, since these type of terms crop up all the time in rotation operators

[@antoncabotta5364]

As a lower 6th student this video blew my brain like no other maths video I have ever seen. This is just crazy.

[@crunk1]

Question: How would one intuit/know if using a matrix as an exponent (as is done in "normal" algebra) would even make sense? Like, how does one make the jump to "let's use the infinite series expansion definition of e^x" and expect it to work for matrices? Does linear algebra have the same properties as "normal" algebra - and therefore we know they can do the same things? Admittedly, I only took one course of Linear Algebra in college and I know next-to-nothing about Abstract Algebra's rings/groups/etc. So, forgive me if my question doesn't make sense.

[@diribigal]

There are a number of other ways to get to exp(A), but one answer to your question is "We don’t have to *expect* the series to make sense in advance. We could just hope it will make sense and then explore it and check."

[@MelindaGreen]

What did this accomplish? He started with e^At for a particular matrix A and proved that it equals e^At which should not be surprising. Given how special that matrix is, this does not seem to be saying anything special about matrices.

[@ernstfrutphlinguhr2494]

Yes, this feels like taking a special case and suggesting that it is fully generisable. He’s so excited to show the connection to calculus that he fails completely to explain the process.

[@MelindaGreen]

@@ernstfrutphlinguhr2494 Yes, who among us hasn't proven that 1 = 1? We usually don't post about it though.

[@elektronikvideos-bremen2873]

And the very last and most beautiful step would be to express -1 as i^2 ❤

[@aisolutionsindia7138]

this was an idea introduced in stochastic differential equations, where youd solve a differential equation by reducing its order and then vectorize it in linear form

[@Baribenatiyacubing]

We got numbers to the power of matrices before GTA 6

[@gregoryfenn1462]

Weird how no one bothered to point out that your matrix is basically i, since AxA is a constant multiple of -I

[@coloneldookie7222]

"The simplest matrix is a 2x2." teeeeeeechnically, it's a 1x1 and we're using square matrices for the identity property, which doesn't work the same for rectangular matrices.

[@Marinealver]

The most complicated thing about matrixes, is when you start putting whole formulas and equations in cells.

[@matesafranka6110]

13:08 My computer graphics brain immediately noticing that as the transformation matrix of 2D rotation

[@javen9693]

The resulting matrix with cos and sin is also a rotation matrix -- basically what computers use to rotate shapes and stuff. so not only is e^matrix a technically possible thing, but it's actually a hugely useful thing that the computer in front of you does millions of times a second every time you lose at video games.

[@rainerzufall42]

I would have defined A as ω * (0, -1; 1, 0)... then you also get the right exponent to ω in the formulas!

[@rainerzufall42]

Additionally, we'd have B² = (0, -1; 1, 0)² = (-1, 0; 0, -1) = - I, so we'd have B² = -1 I as an analogue to i² = -1... resulting in e^(At) = e^(ωt * B) with B² = - I. => e^(At) = e^(B ωt) = I cos(ωt) + B sin(ωt). In German, we'd use E=I and I=B, that's even nicer: e^(I ωt) = E cos(ωt) + I sin(ωt) ... which should remind you of ... e^(i ωt) = cos(ωt) + i sin(ωt).

[@mynameiscian8D]

Even better would be to omit ω entirely, seeing as t is already there. Can easily replace t with ωt or anything else after the calculation!

[@rainerzufall42]

@@mynameiscian8D I had the same idea, but I'm not a physician, so I prefer φ = ωt, which is essentially the same! (φ is the angle that is reached at time t with angle velocity ω) e^(iφ) works as usual.

[@thecakeredux]

I understand how all this is done, but I can't place it anywhere in my mental model of the mathematical world.

[@danielbriggs991]

You can also use it to solve 2x2 systems of linear homogeneous ordinary differential equations with constant coefficients! Arunas Rudvalis was explaining this to me the other day.

[@PaleyDaley]

I am surprised they didn't mention that the final matrix with cos and sin is a 2D rotation matrix.

[@sawelios]

As a side note: that's how we write omega in greek it is usually indistinguishable from English lower case w so don't worry. When we need to use both w and ω we usually do the w with straight lines while the ω with curves

[@ErikScott128]

@@sawelios In school, I had to pretty quickly modify my handwriting to be able to distinguish the symbols when Greek letters started to get introduced. Omega is just one example (I do the same thing, w has straight lines while ω is curved, and I really make the upper parts curl over). The worst are iota and nu, which are almost indistinguishable from i and v if you don't do something significant to distinguish them. Of course, there are letters that are inherently indistinguishable (A and capital Alpha, for example) but thankfully it seems these Greek letters tend to be avoided.

[@shirou9790]

@@ErikScott128 what I do (or was taught to do) is cursive for Latin letters if need be. That way w has a little squiggle that makes it naturally different from ω, and same thing for v and nu. This is also how LaTeX does it--Latin letters are in italics by default in equations, which are similar to cursive. (We also never used iota--just like we'd never use omicron nor capital Alpha for example)

[@joruss]

I feel like if you wrote the first 2 elements as a" x^0/0!" and "x^1/1!" and then simplify it to I and A that'd make more sense for wider audience.

[@gabrielfrank5142]

If you plug in w=pi You get an echo of Euler's formula.

[@hawgokutai]

and the matrix, ([0,-1],[1,0]), is the representation of the imaginary number tying everything together.

[@MooImABunny]

as a physicist, I really love a perspective you get from Lie groups and algebras: it gives an intuitive version of the limit definition of exp(x). when teaching physicists Lie groups, the introduction is of course the 2D rotation matrix, as shown in the video. Since we are expected to already recognize this matrix, so you are instead asked to consider what happens after a tiny time step. A point on the x-axis, rotating clockwise with angular velocity ω has a velocity of (0,ω), so going up in the y-direction, and basically not moving in x. A point on the y-axis meanwhile would have a velocity of (-ω,0), so going left, away from the x-direction. In general, a point (x, y) would have velocity (-ωy, ωx), and since this is linear in the point vector, you describe it with the matrix you gave in the video (vx) = (0 -ω)(x) = ω(0 -1)(x) (vy) (ω 0)(y) (1 0)(y) (btw you can also expand cos(ωt) and sin(ωt), and when you cut off the 2nd power in t and above, you get this same matrix) Now, you are told to imagine this matrix as being a small nudge. This matrix tells every point on the plane where it should go in order to orbit around the origin at angular velocity ω. Of corse, if you tried to use it with a big time step, after a few steps you'd notice you're going off-corse. Say we have ω = π rad/sec, and we took Δt=0.5sec, so we're taking a quarter-turn. Maybe take our point as starting at (1, 0) for simplicity. So we compute it (x1) ~ (x0) + (vx)Δt = ( 1 ) = ( 1 ) (y1) (y0) (vy) (ωΔt) (1.57) this isn't great; we should be completing a quarter-turn, and get (x,y) = (0,1). Okay, let's split Δt into two steps. δt = Δt/2 = 0.25sec (x1) ~ (x0) + (vx0)Δt = ( 1 ) = ( 1 ) (y1) (y0) (vy0)/2 (ωΔt/2) (0.79) (x2) ~ (x1) + (vx1)Δt = ( 1 - (ωΔt/2)^2 ) = (0.38) (y2) (y1) (vy1)/2 ( ωΔt/2 + ωΔt/2 ) (1.57) getting somewhere... so split it into 4 steps. δt = Δt/4 = 0.125sec (x1) ~ (x0) + (vx0)Δt = ( 1 ) = ( 1 ) (y1) (y0) (vy0)/4 (ωΔt/4) (0.39) (x2) ~ (x1) + (vx1)Δt = ( 1 - (ωΔt/4)^2 ) = (0.85) (y2) (y1) (vy1)/4 ( ωΔt/4 + ωΔt/4 ) (0.79) this is getting pretty long, but to make things short, (x4) = (1-(3/8)(ωΔt)^2+(1/16)(ωΔt)^4 -[ωΔt - (1/16)(ωΔt)^3] ) (y4) ( ωΔt - (1/16)(ωΔt)^3 1-(3/8)(ωΔt)^2+(1/16)(ωΔt)^4) = (0.0985) (1.3286) that's closer to (0, 1) If we take 8 steps we end up at (0.02295, 1.1631) Now, consider what operation we're doing with matrices. If we take N steps, each steps looks like (x') = (x) + (0 -ω)(x)Δt = [(1 0) + (0 -ω)Δt] (x) (y') (y) (ω 0)(y)/N (0 1) (ω 0)/N (y) so doing this N times, we just need to compute the matrix power [(1 0) + (0 -ω)Δt]^N = (I + AΔt/N)^N (0 1) (ω 0)/N and we need to take the limit as N→∞. This limit looks awefuly familiar; that's the limit definition of the exponential. exp(x) = lim (1 + x/N)^N but adapted for matrices. Thinking about the intuition for the matrix exponential in the problem above, the matrix A tells us how position becomes velocity, but we can't just take the bulk step, we need to split it into segments, each being N times as small, and let them build up to the true result.

[@Tbop3]

We need to talk about the non-euclidian hairstyle of Tom.

[@timmy18135]

Knock, knock, Neo! It's A Number to the Power of a Matrix!

[@dermaniac5205]

Mind blown when I recognized that the result was a rotation matrix

[@tumultuouscornucopia]

I was all set for him to wrap up with AC theory or transmission lines or computer graphics. But of course this is math - real world applications are beneath them. I liked that the last few seconds alluded to the one fishy bit of being able to take A ^ n. It is different whether A ^ n == A ^ (n-1) * A or A * A ^ (n-1) (although maybe not with his nicely behaved diagonal matrix - exercise left for the reader)

[@ilprincipe8094]

A^n = A^(n-1) * A = A * A^(n-1) is always true for square matrices A, so A^n is perfectly fine. What he is referring to is that in general, AB ≠ BA for A,B some matrices

[@tumultuouscornucopia]

@@ilprincipe8094 right - you've made me work it out with a pencil now ! I don't think it is true for "square" matrices, I think maybe you meant to say "diagonal" matrices, where it does seem to be true - at least for 2x2 which is all I could be bothered trying.

[@codnx]

I dont know anything about matrices yet I could follow this really well, this guy is great at explaining

[@ruudh.g.vantol4306]

1= x^0/0! x = x^1/1! (I always wonder why that is never mentioned)

[@SkorjOlafsen]

For real numbers it seems silly, but the I matrix for the n=0 term kind of makes sense here, because to define the exponential function on e.g. matrices you need to pick a definition of zero and one. So using the I matrix makes things clear that we've chose it as the answer for M^0 (though maybe it's the only choice). But even for matrices there's no reason to make a special case for the n=1 term. You see that a lot for well known series, and I've never understood why some find it simpler to needlessly make special cases from the first term or two.

[@ruudh.g.vantol4306]

Maybe it is because of some cultural rule that an expression must always be “normalized”, similar to how 8/24 would appear as 1/3, even where 8/24 would be more relevant. Ok, it also avoids having to explain why 0! is defined as 1.

[@simeonsurfer5868]

also, the matrix he use can be used in a definition of the complexe number using matrcices, a+ib=aI+bA, where w is 1 in A, and as we have seen in the video the exponentiation of e^(iw) with this definition is respected with this definition.

[@kubilayaytemiz7274]

For omega = i = sqrt(-1), this is the Pauli-Y matrix or gate, which is one of the roots of 2x2 identity matrix and it is widely used in quantum mechanics. Also, exp(At) is the 2D rotation matrix, you can simply multiply any 2D vector by this and it will rotate omega*t radians exactly in the space.

[@ryanvanderpol7962]

Yesssss - so cool, freaking love it! Thanks for making and sharing!!

[@Randomguy-sl8ne]

As a mathematics major i can easily confirm this is most basic stuff

[@tashriquekarriem8865]

Great now do a matrix to the power of matrix 😅

[@jacobs9474]

2x2 Matricies can also represent complex numbers with the matrix multiplication representing stretching rotations, 2x2 quaternions and so on for other fields

[@joaquinclavijo7052]

Just came home from my differential equations exam and saw the notification, I knew this topic very well but it wasn't on the paper...

[@lateAutumn314]

I remember that this was the wildest bit I encountered in my mechanical engineering studies and I loved it.

[@robertolson7304]

E=MC2 or M= area C= circumference 2= parallel. Area is how much. Circumference is how fast (fluidity). Squared is to make it a parallelogram. It will not work for things not parallelograms. Square-able. So someone making something more and more square-able would require some inside out to outside in stuff. Some type of action shape.

[@LeoDaBest_180_Returns]

Amazing as always.

[@manuderezzo]

It does have a nice name. That's the Y Pauli matrix multiplied by -i

[@bsharpmajorscale]

I know from 3B1B something raised to a weird power generally turn out to be a transformation on the 2D plane, and has a cool visualization.

[@bobtk2352]

I would like to see a follow-up video in which the base is not a special number like e. How about 5, for example.

[@ean_596]

Did we actually answer the question though? Doesn't feel like it, but maybe this lets us peel back all the extra conditions added at the beginning? Like, is the series expansion of e even defined for non-square matrices?

[@IrishEye]

Beautiful demonstration in this video.

[@AidenOcelot]

I remember as a kid seeing that matrix multiplication was completely different, and wondered what matrix division would be. I never thought about number-matrix powers

[@qugart.]

This might be the explantion why only the original Matrix movie was good.

[@regulus2033]

12:15 "It doesn't have a nice name" Pauli: "Am I a joke to you????"

[@sciencesdy8793]

13:03 Is also the matrix for rotation transform.

[@Lolwutdesu9000]

Ok but there are a few things missing here: 1) why would you ever raise something to the power of a matrix? What sort of situations demand this? 2) the example provided was awfully convenient in terms of its simplification. Why not try something else like x^A? or pi^A?

[@davidkulmaczewski4911]

In answer to #2... you can always write any value such as "x" or "pi" as a natural exponential. So "x" becomes "e^ln(x), and "pi" becomes e^ln(pi). Because raising a power to a power is the same as multiplying the exponents, x^A becomes (e^ln(x))^A or e^(ln(x)*A), and pi^A becomes (e^ln(pi))^A or e^(ln(pi)*A). Assuming x and pi are constants, this just means multiplying the matrix members by that constant and solving the new equations as shown. In answer to #1.... it's been a while, but I'm pretty sure this would be required (or at least helpful) to solve many advanced differential equations that represent physical systems.

[@bruno_523]

oh, cool, now my nightmares comes to me in youtube video form

[@simongreen9862]

That's amazing! Although I wasn't super shocked to see it work out that way given the e^ix = cos x + isin x relationship. More of a "....oooooooh of course!" kind of amazed... ❤

[@Kneesurgeryistommorrow]

The guy who predicted 301 vies 12 years

[@dougdimmedome5552]

The study of the integral curves of left invariant vector fields on a Lie group.

[@ilprincipe8094]

For some reason i can click on the words "invariant" and "vector fields" in your comment, if I click on them i end up on the yt search page of the respective words. I saw this on a bunch of comments already, and it is very random. New youtube feature?

[@ffggddss]

That antisymmetric matrix, [[0,-1],[1,0]] is often just called, J. It is the generator of the elements of SO(2), i.e., 2-D ("proper," i.e., non-reflection) rotations. There are corresponding matrices J[j,k] in the Special Orthogonal groups, SO(n), in n dimensions, and the relations Tom shows here, allow these rotation groups to be worked with, using these exponentials of matrices. Fred

[@giggen7247]

My first thought when I saw the title is that that can’t be legal.

[@darrelmasterson5850]

it's so strange seeing someone fashioned as a 16 year old emo kid except they have sunken in eyes from old age

[@JaykTheJackal]

It's not too surprising I think that A in this case acts as the imaginary number in euler's formula. That's basically what [ 0, -1; 1, 0 ] does, anyway.

[@TheJuicyTangerine]

I can't not read that exponent as "e to the power of owo"

[@yablaker]

One other crazy fact was missing, that the matrix before sin ((0,-1),(1,0)) is actually an imaginary i. If you square it you get negative Identity matrix. So if you want a nice name for it, how about imaginary Identity (iI)?

[@TerabyteForever]

That was literally unexpected!

[@FASTFASTmusic]

When would you use this? In describing something that is oscillating? Flip-flopping?

[@MuffinsAPlenty]

If you have a system of first order linear ordinary differential equations, this is an effective method of solving that system. (This is what Tom was getting at with the "doing calculus with matrices" stuff)

[@FASTFASTmusic]

@MuffinsAPlenty sorry I wrote it before the end of the video! So it describes a sin wave. I look at these things all the time!

[@gabrielfrank5142]

I wonder if we can use this construct to show there are no "3D" conplex number (or that all higher than 2d extensions of complex numbers must have a 2^n dimensions)