a quaternion version of Euler's formula 

I think you should read the description.

I really liked this video because of all the math this dude did. what a guy

regarding the end, there: it feels as though we'd need to define a separate left-sided exponentiation to account for the lack of commutativity, at a bare minimum

What a gnarly way to start the week. Thank you, professor.

Fascinating, especially the last result which I hadn't seen before. I find that an easy way to grasp quaternionic behavior under exponentiation is to see that any single quaternion (other than a purely real one) defines a complex sub-algebra of the quaternions and that sub-algebra behaves the same way as the complex numbers; its purely imaginary part, unitized, plays the same role as the complex number i. A lot of unintuitive stuff got much simpler when I understood that.

This is the first time quaternions made any sense to me, in spite of several attemps. As ususal presented in a clear no-nonsense way. Thanks a lot!❤

wow, never though that i^j would be equal to i*j=k, thats is cool. Nice video as always Michael! :D

It is not clear to me why i^j should be k instead of -k. Why do we say that (e^(i*pi/2))^j is e^(ij*pi/2) instead of e^(ji*pi/2)?

I am waiting to see the octonions video with the nonassocietivity hell 😂

I loved this! Would love to see this extended to the Clifford representation.

Both of those formulae are special cases of the Clifford algebra formulation of 3D rotations :)

The reason we use H for the quarternions is only secondarily because the letter H stands for Hamilton. Firstly: we can't use the now obvs is Q because that's already taken. This is, of course, a rational reason.

Fun video. Lord Kelvin declared that quaternions, “though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell” 👿

God I love mathematics. It's so lovely to play with ideas in this way.

We're nudging closer to the Clifford Algebra and Geometric Algebra

when I learned that Pauli matrices have such a similar Euler's identity I was really surprised. But if you add the identity you get exactly the quaternion algebra, so knowing that, the quaternion exponentiation makes perfect sense

The title for this video is quite clear and descriptive. The title card is also quite clear and descriptive. Thank you.

Fun video. BTW, the matrix representation of the quaternions is related to the groups SU(2) and its covering group SO(3), and thus they can be used to represent rotations in three dimensions.

Exponentiation of a matrix is just applying the series representation of the exponential to the matrix. It is so amazing to see the concept of exponentials generalize this way... Same for the logarithm. Thank you for making the video!!

Lovely video, as usual. Great channel to watch. Cheers.

My first introduction to quaternions was glancing through an applied electrical engineering text. Fascinating something so theoretical is so essential to describe electrical behavior.

As a physicist I feel like you should discuss the connection to SU(2). It's quite simple to show the version of Euler's formula in that context as well.

This was a very interesting, fun presentation. Thank you professor))

I actually finally understand quaterions now.

I was introduced to quaternions by the video done by 3Blue1Brown... and they keep getting stranger and more interesting with every new thing I learn about them. The exponential transformations exposed in this video are an especially brain-bending result.

That description though lol

I really liked this video because of all the math this dude did. what a guy :)

What are the connection between the quaternions and the Pauli matrices?

I really loved the video it's very good . Thank you sir😊 we will support you

I love the way you present the ideas. Engaging and clear. You asked about the red-brown chalk once. It is a bit hard to see, for example at 2:06. around "example calculations". But these are quibbles. It works for boxes and lines, and the rest of the boards are clear to follow.

Consistency with the order of the imaginary and its coefficient might be a good idea, especially when you start getting into quaternions. I mean, if you try to go the route of complexifying a complex number, (a + bi) + (c + di)j = a + bi + cj + dk is fine but (a + ib) + j(c + id) = a + ib + jc - kd is slightly problematic. You could do (a + ib) + j(c - id) = a + ib + jc + kd, of course, which is a nice callback to the particular complex matrix form you use; though I prefer a different form. Anyway, the signatures are pretty arbitrary here (as sign is, generally) so long as you are consistent, throughout. I'd just stick with trailing imaginaries, personally, but then you run into a formatting issue with Euler's formula. If you want to use the leading i in Euler's and the conjugate imaginary component in the matrix it's probably best to stick with leading everywhere to avoid problems.

Wonderful! Thanks so much 😮

There's a Quarternion version? Damn, you learn something new everyday.

For pedagogical reasons I understand why you left out the multivaluedness of various results, but when it comes to H ≃ M_2(C) matrices being brought to powers of each other, that has A LOT of multivaluedness going on! Log comes with all integer multiples of 2πi times identity matrix.

i^j = ±k depending on how you apply rule for the exponential of an exponential (right vs left).

This is exactly what I needed!!

Great explanation of a tough topic. Great Job!

thanks for your persistence

Quaternions written as matrices were the theme of French most difficult math contest (X-ENS, Maths A) last week, to enter the best engineering school.

I did not expect spinors to make a surprise appearance at the end. But you are spinning things so I guess I should've

Suddenly I feel thankful for Heaviside's version of Maxwell's equations

More Quaternions!

Really interesting, thanks!

I bearly understand normal math and then there is this

I think the contradiction showed in the short video can also be seen here. when you take i^j=(e^{i*pi/2})^j pulling j inside of the exponential isn't well defined I think, would it necessarily be multiplication from the left or should it be from the right? because of the commutation relations i and j have it can be either e^{k*pi/2} or e^{-k*pi/2}.

Some years ago I calculated a general formula for a quaternion raised to a quaternion power. Quite messy and arduos, but was fun and the result is quite neat, a scaled rotation along the axis of both original quations followed by a rotation about the axis of their cross product. (Everytime I say "the axis of" I mean the axis defined by the purely imaginary part, and the cross product being within the imaginary parts also) Edit: of course, this was all abusing notation and glossing over the sketchiness of using exponentials on a non-commutative algebra, but I was on my first year of college so chill I didn't have the knowledge to take that much into consideration xD

When teaching the addition of many disparate components, I like to underline the components as I deal with them. This establishes a bookkeeping methodology for the students (and me!) that prevents losing track of terms in a long list.

What is this video description lol Great vid, I love seeing quaternions at the zoo like this. Wish I saw more in the wild...maybe.

interesting ! thank you very much

13:45 ... My heart sank as I expected you to say " and this would be a nice place to stop" I watched to the end and to be honest, didn't need the matrix stuff, I was already knocked out by the beauty of it all. I dare some elerctrickery wiz will find some use for 'i' and 'k' in their modelling.

for the exercise at 5:27, writing q for a + bi + cj + dk and q*/|q|^2 for (a - bi - cj - dk)/(a^2 + b^2 + c^2 + d^2), one needs to check _both_ q(q*/|q|^2) = 1 and (q*/|q|^2)q = 1, since quaternions are noncommutative - we have |q| ≠ 0 if and only if q ≠ 0 so q*/|q|^2 gives the inverse of any nonzero quaternion, and one can clear the denominators in the checks, meaning it is equivalent to check that qq* = |q|^2 = q*q.

Professor Penn, Why do I think you are a genius?

18:45 Why is it log(A)*B rather than e.g. B*log(A)? Isn't that rather arbitrary choice (unless the matrices commute)?

Very very interesting video.Million thanks from Morocco.

THE DESCRIPTION LMAO

i suggest you start with q=a+b1*i+b2*j+b3*k and b=sqrt(b1^2+b2^2+b3^2) then q=a+b*Î with î the specific normalized quarternion here with |î|=1 and î=b1/b*i+b2/b*j+b3/b*k, then e^q becomes very simular to the complex case: q=a+b*î and e^q=e^a*(cos(b)+sin(b)*î) which notation seems much more alike that in the complex situation. Right?

that was really helpful

I came up with a better, simpler and easier to understand implementation of Euler's formula in quaternion form that can be implemented directly into a computer program to do rotations about an arbitrary axis in 3D space. @FractalWoman "Demystifying Sir William Rowan Hamilton's Quaternions" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-vXyNA0ORYfA.htmlsi=62Eao0gdh7O5LPRp A link to the computer code is available in the description.

interesting that the complex numbers can't just be treated as a plane within the quarternions, but I guess that kind of makes sense the same way that imaginary numbers can't be treated like a number line within the complex numbers

Quaternions, octonions, tetrads, tensors, twistors and vectors are devices for physical description of movement and variation of quantities. This is STRICT computer graphics. Hamilton among many other mathematicians are geniuses.

I guess the description is a mathematical secret code

Hi Michael, there is a major simplification when applying the Power Rule for Exponents to the quaternions, you said exp(i * pi /2) ^ j is exp(i * j * pi / 2) yet I think j being on the right side of i is not so obvious. Why is this the case? Since normally this rule was applied to commutative numbers, now with quaternions there must be a special explanation of this specific order. Edit: Probably this is related to the problem you mention in the short video.

Nice can you do maxwells equations next?!

Gently press

Quaternions' didn't click for me until I learned a little geometric algebra.

1:09 The 'where' could/should include ijk = -1. Then the 6 relations below that follow from these 4 relations, which are easier to remember.

The quatirion field post in exam MathA 2023 in France is beautiful exam to more understanding the quaternions field specialy her matrix represtation

I really like this video this guy is awesome CHALK CHALK CHALK BLACK BOARD

Plotting the unit circle we see that the form given in this video is related to the identity cos squared plus sin squared = 1 by multiplying by the complex conjugate

13:41 this is in fact the quaternionic version of this formula

Equals evil in biome

Do you know the Dihedrons? ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-lqH4BLHGsFw.html What should be the Euler formula for them?

Is this an example of the exponential map from a Lie algebra to its Lie group? The group being the multiplicative group of quaternions of magnitude 1 (so-called versors, isomorphic to SU(2)) and the algebra being the infinitesimal displacements from the identity element, q=1. I'd be grateful for a reply though I'm not sure if I'd understand it -- despite Michael's efforts, Lie algebras still turn my brain to jelly!

I would love to thank you a lot for the video. I have a question (extremely curious): Is that method applicable for the higher Hypercomplex Numbers such as Octonions and Sedenions or there will be some differences in the "B" and "I" construction?

This is quite an intuitive reconstruction of Euler’s formula. But after watching, I’m left to wonder: what about infinite tetrations of quaternions? Are there any that converge to a real and/or transcendental number? And what about the reciprocal of an infinite tetration of quaternions?

nicenice nice i love quaternions :o

Seems like people are very interested in algebraic structures

How much of Complex analysis can be extended to the Quaternions? I have often wondered about this. Do you have any sources or references? Using i as quaternion and complex in same equation is VERY confusing.

You can likewise show that j^i = 1/k, which is pretty cool.

Well, it doesn't commute, I was waiting for that... It's linear transformations, I use those tools a lot for 3D cg

This comes from Clifford algebra, or as it's known these days, geometric algebra.

Neat!

At the 19:42 mark, shouldn't the lower left "i" be "-i" ? In other words, shouldn't the lower left of your matrix "-c+di actually be -(c+di) = -c-di. When you do this, then you get the matrix [0 c+di], [-c-di 0] and when you let c=0 you get [0 i], [-i 0] = k. I can see this clearly when I use the 4x4 matrix implementation of quaternions directly which I show in my video: @FractalWoman "Demystifying Sir William Rowan Hamilton's Quaternions" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-vXyNA0ORYfA.htmlsi=62Eao0gdh7O5LPRp

Editor note: @5:26 - don't use red chalk for text, just outline. Can't read "exercise". Also, it seems there is a mild glare in the center of your board. Change to position lights or camera to not glare.

Question: can you take the logarithm of a matrix by using the spectral decomposition? I've never seen that done but it seems to me that if you can use the spectral decomposition to exponentiate a matrix then it should also be possible to take a matrix and use the spectral decomposition to take its logarithm.

The ending is a bit concerning, as you mentioned. We can still define exp(a+BꞮ) = e^a(cos(B)+Ɪsin(B)). From here, let's define the modulus of a quaternion q = a+bi+cj+dk as |q| = sqrt(a^2+b^2+c^2+d^2). Then, letting q = a+bi+cj+dk = a+BꞮ as above, we can define a quaternionic logarithm as log(q) = ln|q| + Ɪarccos(a/|q|), for some choice of arccos(a/|q|) Then we can check that exp(log(q)) = q for all quaternions q. Next, given two quaternions p and q, the next question is how to define q^p. The sneaky part here is that we have two reasonable options: exp(log(q)*p) or exp(p*log(q)). Note that these two do _not_ have to be equal to each other since quaternion multiplication is noncommutative. My proposal, then, is to define two versions of exponentiation - right exponentiation and left exponentiation. At the risk of confusing with tetration, we use the following notations: qᵖ = "q right exponentiated by p" = exp(log(q)*p) ᵖq = "q left exponentiated by p" = exp(p*log(q)) So in your example of i raised to the j, we have to consider a few things: For q = i, we have a = 0, B = 1, Ɪ = i, |q| = 1, so log(i) = ln(1)+i*arccos(0) = iπ/2, let's say. iʲ = "i right exponentiated by j" = exp(log(i)*j) = exp(ijπ/2) = exp(kπ/2) = cos(π/2)+k*sin(π/2) = k ʲi = "i left exponentiated by j" = exp(j*log(i)) = exp(jiπ/2) = exp(−kπ/2) = cos(−π/2)+k*sin(−π/2) = −k This explains the discrepancy in your short where you talk about this. Some interesting things to point out here: if r is a positive real number, then log(r) = ln|r|, which is a real number and commutes with all quaternions. So for any quaternion p and positive real number r, we have rᵖ = ᵖr (i.e., right and left exponentiation of r by q produces the same result). So we haven't introduced an incompatibility between e^p and exp(p) (using a good choice of argument).

I dont know why he said i^i is such as famous video concept. That has to be one of the easiest math results to arrive at that I can thing of...

Wonderful video but I haven't understood the last step, could you please explain or prove it .

Any chance of getting a link to the video you're talking about that explains why this definition is sketchy? :)

Quaternions remind me a lot of the pseudo-Euclidean geometry of special relativity. One "real" component represents time, and three "imaginary" ones represent space.

At 12:38 that seems to be to be a huge leap of faith. I expect that if you, say, expanded e^x to a series, stuck I in it, and simplified, I can see you could get the stated result, but saying it is so just because I^2=-1 seems like an awful stretch.

Can we have nonstandard quaternions next, please?

I have to say that I also was very suspicious of the exponential power equation. In effect, you want to find a way that given an analytic function of two complex variables f(x,y), you can extend it to a non-commutative algebra f(A,B). I think that the problems you found show that this can't be done in a unique way. For example with matrices, the power law exp(A+B) = exp(A)exp(B) is not true. So you cannot simply assume that works in the commutative case works in the non-commutative case.

Can jAe^jB((iCe^iD) - (iEe^iF)) be put in the form I*Ge^I*H, where i is complex, j is part of the quaternions, and I is the same as your notation?

Hi Dr. Penn! I think the intro is a little too loud.

But you didn't show *why*, just because the quaternion I has the property I² = -1, you can apply the Euler formula as if it were a complex number

Does the entierty of the boards after 13:48 belong to : e^q = e ^a ( .... and there is no closing parenthesis.

I legit wanna know where the inspiration for the descriptions come from :D

Wait so I know i = e^i * pi/2 and j and k but what aboit a general quaternion like i+2j+k? What is that in exponential form?

So what if you have an infinite expression of square root (i) with square root (j) inside of that with square root (k) inside of that and so on and then took the limit of that expression? Surely there’s a mathstack about that!