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 🙏🏻
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)
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'.
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
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.
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.
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 RUclips has some excellent lectures on this stuff, and it's honestly a much better introduction to differential equations than what I had in undergrad.
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.
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.
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.
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.
I took my linear algebra final yesterday, and what's weird to me is how _simple_ all of the steps taken in this video seemed to me. I was nodding along the whole time, going "yeah, that makes sense, I completely understand how he got from here to there." I'd be surprised where he went with it, like at the end when he linked it to a differential equation, but even then I completely understood the step he took once he showed it. It's a weird feeling, especially knowing a few years ago I wouldn't have understood any part of this video at all...
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.
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.
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
Learning the pure maths behind its applications is always fun. I first learned about this in quantum mechanics and thought "what the heck is this wizardry?" 😄
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.
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.
I didn't understand, but I got that feeling that I will remember and rewatch this video over and over again somewhere in the future, when I study something related to this in college. It's like I discovered some part of my brain that I haven't unlocked yet.
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.
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.
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
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
Месяц назад+24
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.
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.
This is why I like mathematics, you can use different ways to get to an answer and it still connects to previous concepts. Even when applied to something wild and new like this
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!!
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.
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.
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.
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.
@@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?
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
@@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.
@@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)
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.
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.
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.
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."
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.
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
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
2x2 Matricies can also represent complex numbers with the matrix multiplication representing stretching rotations, 2x2 quaternions and so on for other fields
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
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 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.
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?
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.
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.
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.
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.
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
Yay! Complex numbers rediscovered in their true form. After all, multiplying by a complex number is supposed to scale/rotate a complex number, so no surprise they end up being 2x2 matrices.
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.
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)
I was at school for the tail end of "modern maths", so we learned about vectors and matrices before we learned about algebra. Turns out we were skipping straight to calculus.
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?
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.
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... ❤
But... looking it from the algebra side; what does exactly Number to the Power of a Matrix mean? How many times are we multiplying E exactly? does the question makes sense? Because that would mean that there are "matrix roots" of numbers. I gues is the same as imaginary numbers, I get they are "rotations", but I always felt there must be a way to explain it so it makes sense from the "multiplication of the same number n-times" point of view
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)
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
@@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.
I have a vague memory from advanced calculus taken 50 years ago, proofing this Identity as a Taylor Series. This has Leonard Euler written all over it.
i hope numberphile makes more geometric algebra videos and how it simplifies calculations. you don't necessarily need linear algebra if you use geometric algebra. so much easier. i don't know why it wasn't taught first
This is useful. Replace t with Qs(my symbol for quantum entropy). Thus the change in the quantum field matrix(z) with respect to change in quantum entropy: dz/dQs this is a measure of the squeeze of the quantum field.
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.
Coming soon - a number to the power of an operator... oh hello Quantum Mechanics.
just exponentiate hamiltonians and be happy, physicists are insane
Every good programmer knows to keep data and code separated. 😆
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!
@@sirusdesnoes3094totally agree with you
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 🙏🏻
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)
nice
coming soon to a numberphile near you
The relation of complex numbers here also makes sense considering the resulting matrix of e^A is also just the 2-dimensional rotation matrix!
Very well pointed
i noticed where it was going as soon as he said "(0, -w, w, 0)"
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'.
And shows up in Electrical Engineering (Alternating Current etc).
And also in 3D computer graphics
Exactly! 👍😎
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
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.
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.
Was going to mention it too!
omega radians ;)
@@HyperCubist Actually, radians/second, with t as time in seconds.
Or with t in any time units, and ωt in radians.
Fred
So that's why it seemed so familiar.
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 RUclips has some excellent lectures on this stuff, and it's honestly a much better introduction to differential equations than what I had in undergrad.
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.
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.
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.
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.
I took my linear algebra final yesterday, and what's weird to me is how _simple_ all of the steps taken in this video seemed to me. I was nodding along the whole time, going "yeah, that makes sense, I completely understand how he got from here to there." I'd be surprised where he went with it, like at the end when he linked it to a differential equation, but even then I completely understood the step he took once he showed it. It's a weird feeling, especially knowing a few years ago I wouldn't have understood any part of this video at all...
Even after 13 years of watching, the videos still amaze me. Fantastic work and effort as always, thanks for sharing with us!
OMG this was crazy, really nice when you can go from linear algebra to calculus without expecting it.
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.
Rumor has it that famed theoretical physicist Paul A. M. Dirac first figured this out, in his head, while attending a cocktail party.
I'll have what he (Dirac) is having.
My favorite physicist
The flow and clarity of Tom's lectures have improved immensely.
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.
"A number to the power of a matrix" sounds like "coloring a moral compass with ice cream flavors" to me.
Indeed, and if you ever find a way to make operational sense of that proposition, I'm all in!
@@jespervalgreen6461graphics programming :)
Well can you do a reman sum where n =/= an integer (i.e. 2.5).
And the craziest part is we actually use this very regularly! It came up on my graduate quantum mechanics final :)
cry about it
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
Learning the pure maths behind its applications is always fun. I first learned about this in quantum mechanics and thought "what the heck is this wizardry?" 😄
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.
12:15 "It doesn't have a nice name"
Pauli: "Am I a joke to you????"
As a lower 6th student this video blew my brain like no other maths video I have ever seen. This is just crazy.
Math has endless cool things like this the more you study it
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.
I didn't understand, but I got that feeling that I will remember and rewatch this video over and over again somewhere in the future, when I study something related to this in college. It's like I discovered some part of my brain that I haven't unlocked yet.
The best numberphile video I have seen in a long time.
As an Electrical Engineer, I feel right at home in this video 🤣
Was about to write something along the same line. Physics for the win !
As an electrical engineer I am also the inventor of Imaginary (Complex) Chess, the most complex chess variant ever invented.
EE151 Circuit Theory ☺
I can feel your pain bro.
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.
Class 😎👍🏻
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.
The final calculus "trick" was just amazing
Matrix Exponential was something that was drilled into us during grad school engineering.
instead of grad school I went straight into the field, so I missed out on a lot of the cooler math concepts 😭
@@LonkinPork quantum field theory or normal field theory?
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
Not factoring out the omega for so long was diabolical, it had me on the edge of my seat.
just sat my last final before the break.. somehow toms convinced me i still need MORE math. cheers and happy holidays!
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.
Shhh you'll have people smashing all their favourite toys together soon
This stuff in this video turns out to be super important in physics. They play these games and useful tools come out, somehow.
Not as difficult as the math in chaos theory or nullo space !
I remember encountering it for the first time in quantum mechanics course, i was so confused haha
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.
What an incredible video! Really shows the deep connections that exist between different branches of mathematics.
"Why the tea?" British people always need tea.
You forgot to mention that the rotation matrix magically pops out, which is really cool in my opinion.
I dont know anything about matrices yet I could follow this really well, this guy is great at explaining
I love to play with numbers in mathematics.
There is another level of enjoyment by solving these types of problems.
This is why I like mathematics, you can use different ways to get to an answer and it still connects to previous concepts. Even when applied to something wild and new like this
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!!
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.
And the very last and most beautiful step would be to express -1 as i^2 ❤
I love that he also derived Euler's formula with the matrix definition of i.
Still struggling to understand how or why this would be a valuable operation, but trying to think about it is extremely interesting
Basically this is how the math works for the Heisenberg picture of quantum mechanics (extremely useful)
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.
@@javen9693 Although Woody probably uses quaternions.
Computer Science and Quantum Physics. And if Crypto-currencies gain liquidity and become legal tender, the financial sector as well.
It seems A is a generator for rotations in the plane. Could you explain the relationship with Lie algebra and Lie group?
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.
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.
Yesssss - so cool, freaking love it! Thanks for making and sharing!!
What's next, a matrix root?
Matrix factorial?
Once you have diagonalized the matrix, it is all easy.
@@landsgevaer thatis, if your matrix is diagonalizable
You can always use continuity though! The diagonalisable matrices are dense in the set of all matrices!
@@AgentM124matrix square root can be defined more generally with jordan decomposition and the cauchy integral formula
The factorial of any square matrix with real or complex coefficients is in fact the null matrix. See also "nilpotent".
13:08 My computer graphics brain immediately noticing that as the transformation matrix of 2D rotation
Its gonna equal pi, isn't it.
I mean, a matrix can't equal pi since it's a scalar, but, it's gonna be something related, calling it.
πI you mean?
@@quinnencrawford9707no.
@@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?
Or wau
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
@@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.
@@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)
I remember that this was the wildest bit I encountered in my mechanical engineering studies and I loved it.
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.
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.
@@ernstfrutphlinguhr2494 Yes, who among us hasn't proven that 1 = 1? We usually don't post about it though.
Beautiful demonstration in this video.
Amazing as always.
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.
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."
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...
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.
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
Mind blown when I recognized that the result was a rotation matrix
It does have a nice name. That's the Y Pauli matrix multiplied by -i
There are some of these videos where I'm like "oh I get it. I learned something today" and then there's what we have here today.
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
2x2 Matricies can also represent complex numbers with the matrix multiplication representing stretching rotations, 2x2 quaternions and so on for other fields
I'm glad I spent five years doing mechanical engineering just so I can understand a Numberphile video 25 years later. 😂
We got numbers to the power of matrices before GTA 6
Weird how no one bothered to point out that your matrix is basically i, since AxA is a constant multiple of -I
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
I would have defined A as ω * (0, -1; 1, 0)... then you also get the right exponent to ω in the formulas!
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).
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!
@@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.
The study of the integral curves of left invariant vector fields on a Lie group.
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?
Best numberphile video ever
We actually explored this idea in Linear Algebra 1 (!) in our first semester of mathematics, absolutely bonkers.
I am surprised they didn't mention that the final matrix with cos and sin is a 2D rotation matrix.
My first thought when I saw the title is that that can’t be legal.
1= x^0/0!
x = x^1/1!
(I always wonder why that is never mentioned)
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.
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.
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.
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.
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
Yay! Complex numbers rediscovered in their true form. After all, multiplying by a complex number is supposed to scale/rotate a complex number, so no surprise they end up being 2x2 matrices.
The most complicated thing about matrixes, is when you start putting whole formulas and equations in cells.
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.
When would you use this? In describing something that is oscillating? Flip-flopping?
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)
@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!
I was at school for the tail end of "modern maths", so we learned about vectors and matrices before we learned about algebra. Turns out we were skipping straight to calculus.
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?
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.
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... ❤
I should go back to Mathematics. This is fun.
If you want to know more look up 'functional calculus' and the spectral theorem of which this e^A is an example.
I encountered this in masters level aircraft dynamics... mind blown, never feared higher math again.
But... looking it from the algebra side; what does exactly Number to the Power of a Matrix mean? How many times are we multiplying E exactly? does the question makes sense?
Because that would mean that there are "matrix roots" of numbers. I gues is the same as imaginary numbers, I get they are "rotations", but I always felt there must be a way to explain it so it makes sense from the "multiplication of the same number n-times" point of view
13:20 "What was the job of the t?" "Oh it's just cause we're British *takes a sip*"
I have only just got my head around group theory. To the power of a matrix sounds like tricky maths.
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)
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
@@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.
Lovely video!!! The "scratching" sound of marker on paper😭
I have a vague memory from advanced calculus taken 50 years ago, proofing this Identity as a Taylor Series. This has Leonard Euler written all over it.
i hope numberphile makes more geometric algebra videos and how it simplifies calculations. you don't necessarily need linear algebra if you use geometric algebra. so much easier. i don't know why it wasn't taught first
Seems rather familiar... I think we did a similar derivation in my electrical engineering classes back in college.
This is useful. Replace t with Qs(my symbol for quantum entropy). Thus the change in the quantum field matrix(z) with respect to change in quantum entropy: dz/dQs this is a measure of the squeeze of the quantum field.
Tom is so nice to look at, but as usual, I am instantly lost, lol.
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.
It looks like doing this produces a rotation matrix, neat
and the matrix, ([0,-1],[1,0]), is the representation of the imaginary number tying everything together.
If you set t=pi and w=1, it evaluates to -I!