How (and why) to raise e to the power of a matrix | DE6

General exponentials, love, Schrödinger, and more. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: https://3b1b.co/mat-exp-thanks Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ The Romeo-Juliet example is based on this essay by Steven Strogatz: http://www.stevenstrogatz.com/essays/loves-me-loves-me-not-do-the-math The book shown at the start is Vladimir Arnold's (excellent) textbook on ordinary differential equations. https://amzn.to/3dtXSwj Need a review of ordinary powers of e? https://youtu.be/m2MIpDrF7Es Or of linear algebra? https://youtu.be/kYB8IZa5AuE Timetable 0:00 - Definition 6:40 - Dynamics of love 13:17 - Linear systems 20:03 - General rotations 22:11 - Visualizing with flow ------------------ Code for this video: https://github.com/3b1b/videos/blob/master/_2021/matrix_exp.py These animations made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown