Final version of raising e to the power of a matrix
Added 2021-04-01 18:47:53 +0000 UTCHi everyone,
The final version is out! For those of you who watched the draft, you'll find new content starting around 22:00, explaining the flow visualization and giving the main idea for why exp(Mt) solves the equations we are claiming it does.
The other big structural changes are to have aggressively cut down the section on real-number exponents and to reorder things so that the geometric solutions happens much earlier so that once we get to explicitly computing exp(90-degree-matrix), it's more a matter of corroborating the claim, rather than finding the solution for the first time.
Thanks so much for your feedback on the draft! So many of you had great suggestions, and I hope you understand that I cannot incorporate them all. My own notes for this video include an ungodly number of pages with potential alternative approaches and interesting facts to bring up. Just know that I do find your comments very helpful during final edits.
--
In unrelated news, I've been seeing manim pop up more often in YouTube videos these days. It made a brief appearance in the most recent video by Veritasium, who by the way has been killing it on the math content recently, and continues to be used by Reducible (whom you should all check out), for example in his recent great video on the GJK algorithm.
Much of this is thanks to the ManimCommunity repo making it more friendly to get started with. Still, it gives some kind of nice feeling that when I pour time in the code for visuals on these videos, there's some hope it might be useful beyond what I make.
-Grant
Comments
looking forward to watching the video about e^Mt properties.
2021-12-13 04:27:59 +0000 UTCEsto es unico
2021-06-16 12:55:30 +0000 UTCIt's a miracle to me how people ever became inspired to pursue maths without Grants intervention
2021-04-28 17:33:08 +0000 UTCMy friend told me about you yesterday. This video blew my mind. I hold a BS in math.
2021-04-24 01:07:22 +0000 UTCthey always like to make it formal and complicated in France . i know that because i am french
2021-04-23 18:13:54 +0000 UTCThanks Jason!
3blue1brown
2021-04-18 15:46:06 +0000 UTCI learned quaternions before pauli and was still convinced that physicists used matrices because they understood complex numbers as black boxes instead of as rotations. Just here to support un-black-boxing of geometric concepts behind algebra
Jason Hise
2021-04-16 11:26:43 +0000 UTCHonestly I’m unfamiliar with any applications to confusion matrices. Let me know if you find any of note.
3blue1brown
2021-04-04 16:35:52 +0000 UTCI don't know if anyone else has said this but when we were taught complex numbers in France, they were initially given as matrices (a -b, b a). We weren't allowed to use the a+bi notation until we'd shown that they were equivalent.
Steve
2021-04-03 13:50:47 +0000 UTCOh! It's the Taylor Series. Cool! Did not see that coming. (Though I guess I should have.) e^(a d/dx) f(x) = f(x+a). Wild.
Mike Jarvis
2021-04-03 03:57:53 +0000 UTCI always love your videos, even when I already understand the material. I've actually used matrix exponentiation in my research before, but as always, your visual explanation of it helped clarify things further in my head. But what blew my mind, and why I'm writing, is your final teaser, saying to consider e^(d/dx). (!!!!!) Mind blown. I think I have an inkling of what that might mean, but barely. And absolutely no intuitions about it at all. Please do a video on that!
Mike Jarvis
2021-04-03 03:28:41 +0000 UTCHow are you doing the calculation? That looks like it might just be exponentiating each term of the matrix.
3blue1brown
2021-04-03 02:41:01 +0000 UTCSorry about that, just fixed.
3blue1brown
2021-04-03 02:39:16 +0000 UTCA beautiful video, nevertheless I run immediately in trouble trying to recreate the calculation of e^(A*C) where A=matrix, C=constant. example from your video: A = {{-1, -1}, {1, 0}} and C = 0.27 you give the solution: {{-0.73, -0.23}, {0.23, 0.97}}. I get, in Geogebra: {{0.76, -0.76}, {1.31, 1}}. For sure the wrong calculation is on my side, but I wonder what I am doing wrong. Geogebra command: ℯ^(A C)
2021-04-03 01:46:51 +0000 UTCI’m pondering the application of this when it comes to transforming a contingency table or a confusion matrix. It seems this transform is good for dependent variables with respect to some intervention. It also seems good to measure the extent of the dependency (vector magnitude). Would this also serve as a test for dependence? Or how does it relate to chi squared, etc.?
2021-04-02 14:14:33 +0000 UTCThe two links in the sentence: "Two worth highlighting include this video by Mathemaniac on the Jacobian, and this recent one by Reducible on the GJK algorithm." are actually identical. I'm just curious what the other video is. Can you share the other link here, please? EDIT: Nevermind, I found it: https://youtu.be/ajv46BSqcK4
Jan Hrček
2021-04-02 06:46:10 +0000 UTCIt's one way to fix an issue :)
3blue1brown
2021-04-01 22:15:28 +0000 UTCOh my mistake! Someone had shared that video with me because they saw manim there, I'll edit the post.
3blue1brown
2021-04-01 22:11:04 +0000 UTCI didn't think you'd be able to improve this video significantly, but you proved me wrong! I loved how you ended it by discussing the solutions in terms of vector flow. Also, you fixed the problem of using the wrong image for Goldbach by eliminating this portion of the video. Great job!
David Terr
2021-04-01 20:30:05 +0000 UTC
