title

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

description

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
------------------
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
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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
------------------
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detail

{'title': 'How (and why) to raise e to the power of a matrix | DE6', 'heatmap': [{'end': 622.904, 'start': 585.649, 'weight': 0.903}, {'end': 701.936, 'start': 646.335, 'weight': 0.718}, {'end': 740.591, 'start': 717.236, 'weight': 0.715}, {'end': 804.888, 'start': 780.409, 'weight': 0.771}, {'end': 890.462, 'start': 857.871, 'weight': 0.946}, {'end': 1026.124, 'start': 1006.499, 'weight': 0.797}], 'summary': 'Covers matrix exponentials in differential equations, including their visualization in quantum mechanics and practical applications, such as predicting outcomes in finance and epidemic modeling, with a focus on rotation in state space and its significance in various fields.', 'chapters': [{'end': 83.513, 'segs': [{'end': 83.513, 'src': 'embed', 'start': 22.173, 'weight': 0, 'content': [{'end': 27.077, 'text': 'Now, taken out of context, putting a matrix into an exponent like this probably seems like total nonsense.', 'start': 22.173, 'duration': 4.904}, {'end': 30.479, 'text': 'But what it refers to is an extremely beautiful operation.', 'start': 27.577, 'duration': 2.902}, {'end': 33.402, 'text': "And the reason it shows up in this book is that it's useful.", 'start': 30.94, 'duration': 2.462}, {'end': 37.025, 'text': "It's used to solve a very important class of differential equations.", 'start': 33.822, 'duration': 3.203}, {'end': 42.527, 'text': 'In turn, given that the universe is often written in the language of differential equations.', 'start': 37.765, 'duration': 4.762}, {'end': 48.811, 'text': 'you see this pop up in physics all the time too, especially in quantum mechanics, where matrix exponents are littered throughout the place.', 'start': 42.527, 'duration': 6.284}, {'end': 50.732, 'text': 'They play a particularly prominent role.', 'start': 48.991, 'duration': 1.741}, {'end': 57.815, 'text': "This has a lot to do with Schrodinger's equation, which we'll touch on a bit later, and it may also help in understanding your romantic relationships,", 'start': 51.312, 'duration': 6.503}, {'end': 59.436, 'text': 'but again all in due time.', 'start': 57.815, 'duration': 1.621}, {'end': 75.105, 'text': 'A big part of the reason I want to cover this topic is that there is an extremely nice way to visualize what matrix exponents are actually doing using flow that not a lot of people seem to talk about.', 'start': 65.737, 'duration': 9.368}, {'end': 76.987, 'text': 'But for the bulk of this chapter.', 'start': 75.666, 'duration': 1.321}, {'end': 83.513, 'text': "let's start by laying out what exactly the operation is and see if we can get a feel for what kinds of problems it helps us to solve.", 'start': 76.987, 'duration': 6.526}], 'summary': 'Matrix exponentiation solves differential equations, crucial in physics and quantum mechanics.', 'duration': 61.34, 'max_score': 22.173, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo22173.jpg'}], 'start': 0.189, 'title': 'Matrix exponents in differential equations', 'summary': 'Explores the concept of using matrix exponents to solve differential equations, particularly in the context of quantum mechanics, emphasizing visualization using flow.', 'chapters': [{'end': 83.513, 'start': 0.189, 'title': 'Matrix exponents in differential equations', 'summary': 'Explores the concept of using matrix exponents to solve differential equations, particularly in the context of quantum mechanics, and emphasizes the visualization of this operation using flow.', 'duration': 83.324, 'highlights': ['Matrix exponents are used to solve a very important class of differential equations, which is crucial in understanding the universe and frequently appears in physics, especially in quantum mechanics.', 'The concept of matrix exponents offers an extremely nice way to visualize their operation using flow, providing a unique perspective that is not widely discussed.', "Matrix exponents are employed in understanding Schrodinger's equation, a fundamental principle in quantum mechanics."]}], 'duration': 83.324, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo189.jpg', 'highlights': ['Matrix exponents are used to solve a very important class of differential equations, crucial in understanding the universe and frequently appears in physics, especially in quantum mechanics.', "Matrix exponents are employed in understanding Schrodinger's equation, a fundamental principle in quantum mechanics.", 'The concept of matrix exponents offers an extremely nice way to visualize their operation using flow, providing a unique perspective that is not widely discussed.']}, {'end': 393.947, 'segs': [{'end': 160.204, 'src': 'embed', 'start': 100.739, 'weight': 0, 'content': [{'end': 107.001, 'text': 'For example, if I took the number 2 and plugged it into this polynomial, then, as you add more and more terms,', 'start': 100.739, 'duration': 6.262}, {'end': 122.133, 'text': 'each of which looks like some power of 2 divided by some factorial, The sum approaches a number near 7.389, and this number is precisely e times e.', 'start': 107.001, 'duration': 15.132}, {'end': 127.219, 'text': 'If you increment this input by 1, then somewhat miraculously, no matter where you started from,', 'start': 122.133, 'duration': 5.086}, {'end': 130.842, 'text': 'the effect on the output is always to multiply it by another factor of e.', 'start': 127.219, 'duration': 3.623}, {'end': 137.911, 'text': "For reasons that you're going to see in a bit, mathematicians became interested in plugging all kinds of things into this polynomial.", 'start': 132.305, 'duration': 5.606}, {'end': 141.715, 'text': 'Things like complex numbers, and for our purposes today, matrices.', 'start': 138.251, 'duration': 3.464}, {'end': 145.639, 'text': 'Even when those objects do not immediately make sense as exponents.', 'start': 142.295, 'duration': 3.344}, {'end': 151.96, 'text': 'What some authors do is give this infinite polynomial the name exp when you plug in more exotic inputs.', 'start': 146.618, 'duration': 5.342}, {'end': 157.643, 'text': "It's a gentle nod to the connection that this has to exponential functions in the case of real numbers,", 'start': 152.541, 'duration': 5.102}, {'end': 160.204, 'text': "even though obviously these inputs don't make sense as exponents.", 'start': 157.643, 'duration': 2.561}], 'summary': 'Plugging numbers into a polynomial approaches e times e, and incrementing the input by 1 always multiplies the output by another factor of e.', 'duration': 59.465, 'max_score': 100.739, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo100739.jpg'}, {'end': 227.437, 'src': 'embed', 'start': 200.482, 'weight': 3, 'content': [{'end': 204.725, 'text': 'That way you can multiply it by itself according to the usual rules of matrix multiplication.', 'start': 200.482, 'duration': 4.243}, {'end': 207.227, 'text': 'This is what we mean by squaring it.', 'start': 205.346, 'duration': 1.881}, {'end': 215.614, 'text': 'Similarly, if you were to take that result and then multiply it by the original matrix again, this is what we mean by cubing the matrix.', 'start': 208.088, 'duration': 7.526}, {'end': 221.694, 'text': 'If you carry on like this, you can take any whole number power of a matrix.', 'start': 217.952, 'duration': 3.742}, {'end': 222.815, 'text': "It's perfectly sensible.", 'start': 221.834, 'duration': 0.981}, {'end': 227.437, 'text': 'In this context, powers still mean exactly what you would expect, repeated multiplication.', 'start': 223.335, 'duration': 4.102}], 'summary': 'Matrix can be raised to any whole number power through repeated multiplication.', 'duration': 26.955, 'max_score': 200.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo200482.jpg'}, {'end': 320.652, 'src': 'embed', 'start': 291.175, 'weight': 1, 'content': [{'end': 298.201, 'text': 'And as it keeps going, it seems to be approaching a stable value, which is around negative one times the identity matrix.', 'start': 291.175, 'duration': 7.026}, {'end': 302.083, 'text': 'In this sense, we say the infinite sum equals that negative identity.', 'start': 298.881, 'duration': 3.202}, {'end': 307.426, 'text': 'By the end of this video, my hope is that this particular fact comes to make total sense to you.', 'start': 303.004, 'duration': 4.422}, {'end': 312.329, 'text': "For any of you familiar with Euler's famous identity, this is essentially the matrix version of that.", 'start': 307.946, 'duration': 4.383}, {'end': 320.652, 'text': 'It turns out that, in general, no matter what matrix you start with, as you add more and more terms, you eventually approach some stable value.', 'start': 313.089, 'duration': 7.563}], 'summary': 'The infinite sum approaches a stable value, negative identity matrix. generalizable to all matrices.', 'duration': 29.477, 'max_score': 291.175, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo291175.jpg'}, {'end': 404.449, 'src': 'embed', 'start': 372.3, 'weight': 5, 'content': [{'end': 375.081, 'text': 'The process of discovering math typically goes the other way around.', 'start': 372.3, 'duration': 2.781}, {'end': 380.343, 'text': 'They start by chewing on specific problems and then generalizing those problems,', 'start': 375.522, 'duration': 4.821}, {'end': 383.504, 'text': 'then coming up with constructs that might be helpful in those general cases.', 'start': 380.343, 'duration': 3.161}, {'end': 388.126, 'text': 'And only then do you write down a new definition or extend an old one.', 'start': 384.084, 'duration': 4.042}, {'end': 393.947, 'text': 'As to what sorts of specific examples might motivate matrix exponents, two come to mind.', 'start': 389.426, 'duration': 4.521}, {'end': 397.448, 'text': 'One involving relationships, and the other quantum mechanics.', 'start': 394.507, 'duration': 2.941}, {'end': 399.288, 'text': "Let's start with relationships.", 'start': 398.188, 'duration': 1.1}, {'end': 404.449, 'text': 'Suppose we have two lovers.', 'start': 403.249, 'duration': 1.2}], 'summary': 'Math discovery starts with specific problems, then generalizes, and may be motivated by relationships or quantum mechanics.', 'duration': 32.149, 'max_score': 372.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo372300.jpg'}], 'start': 84.392, 'title': 'Matrix exponentials', 'summary': 'Delves into the definition of matrix exponential, its connection to exponential functions, and practical applications, as well as the concept of raising a matrix to infinite power and its significance, illustrated by an example approaching a stable value of negative one times the identity matrix.', 'chapters': [{'end': 248.472, 'start': 84.392, 'title': 'The exponential of a matrix', 'summary': 'Explains the definition of the exponential of a matrix using a certain infinite polynomial, its connection to exponential functions, and the practical application of this concept in matrix operations.', 'duration': 164.08, 'highlights': ['The exponential of a matrix is defined using an infinite polynomial, known as its Taylor series, which involves adding more terms that approach a number near 7.389, precisely e times e.', 'Plugging different inputs into the infinite polynomial, including complex numbers and matrices, demonstrates the connection to exponential functions and signifies the definition for exotic inputs.', 'The practical application of the exponential of a matrix involves multiplying the matrix by itself, following the rules of matrix multiplication, to obtain powers of the matrix, which represents repeated multiplication.']}, {'end': 393.947, 'start': 249.532, 'title': 'Matrix exponents and infinite sums', 'summary': 'Discusses the concept of raising a matrix to the infinite power, showcasing an example where an infinite sum of a 2x2 matrix approaches a stable value of negative one times the identity matrix, leading to a broader understanding of the significance and applicability of this operation.', 'duration': 144.415, 'highlights': ['An infinite sum of a 2x2 matrix approaches a stable value of negative one times the identity matrix.', 'Matrix exponents present a new operation that requires understanding its significance and predictive capabilities.', 'Mathematicians develop definitions and constructs based on specific problems and generalization, rather than starting from definitions and theorems.']}], 'duration': 309.555, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo84392.jpg', 'highlights': ['The exponential of a matrix is defined using an infinite polynomial, known as its Taylor series, approaching a number near 7.389, precisely e times e.', 'An infinite sum of a 2x2 matrix approaches a stable value of negative one times the identity matrix.', 'Plugging different inputs into the infinite polynomial demonstrates the connection to exponential functions and signifies the definition for exotic inputs.', 'The practical application of the exponential of a matrix involves multiplying the matrix by itself to obtain powers of the matrix, representing repeated multiplication.', 'Matrix exponents present a new operation that requires understanding its significance and predictive capabilities.', 'Mathematicians develop definitions and constructs based on specific problems and generalization, rather than starting from definitions and theorems.']}, {'end': 768.608, 'segs': [{'end': 508.456, 'src': 'embed', 'start': 465.785, 'weight': 0, 'content': [{'end': 472.251, 'text': "As Romeo's love increases in response to Juliet, her equation continues to apply and drives her love down.", 'start': 465.785, 'duration': 6.466}, {'end': 478.356, 'text': 'Both of these equations always apply from each infinitesimal point in time to the next,', 'start': 473.412, 'duration': 4.944}, {'end': 483.18, 'text': 'so every slight change to one value immediately influences the rate of change of the other.', 'start': 478.356, 'duration': 4.824}, {'end': 486.542, 'text': 'This is a system of differential equations.', 'start': 484.301, 'duration': 2.241}, {'end': 494.447, 'text': "It's a puzzle where your challenge is to find explicit functions for x and y that make both of these expressions true.", 'start': 487.143, 'duration': 7.304}, {'end': 498.029, 'text': 'Now, as systems of differential equations go,', 'start': 495.567, 'duration': 2.462}, {'end': 503.652, 'text': 'this one is on the simpler side enough so that many calculus students could probably just guess at an answer.', 'start': 498.029, 'duration': 5.623}, {'end': 508.456, 'text': "But keep in mind, it's not enough to find some pair of functions that makes this true.", 'start': 504.434, 'duration': 4.022}], 'summary': "Romeo's love increases as juliet's decreases in a system of differential equations.", 'duration': 42.671, 'max_score': 465.785, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo465785.jpg'}, {'end': 558.193, 'src': 'embed', 'start': 530.757, 'weight': 1, 'content': [{'end': 533.799, 'text': 'Very often, when you have multiple changing values like this,', 'start': 530.757, 'duration': 3.042}, {'end': 538.101, 'text': "it's helpful to package them together as coordinates of a single point in a higher dimensional space.", 'start': 533.799, 'duration': 4.302}, {'end': 540.242, 'text': 'So for Romeo and Juliet,', 'start': 538.741, 'duration': 1.501}, {'end': 550.187, 'text': "think of their relationship as a point in a 2D space the x-coordinate capturing Juliet's feelings and the y-coordinate capturing Romeo's.", 'start': 540.242, 'duration': 9.945}, {'end': 558.193, 'text': "Sometimes it's helpful to picture this state as an arrow from the origin, other times just as a point.", 'start': 553.348, 'duration': 4.845}], 'summary': "Visualize romeo and juliet's relationship as a 2d point in a higher dimensional space.", 'duration': 27.436, 'max_score': 530.757, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo530757.jpg'}, {'end': 622.904, 'src': 'heatmap', 'start': 585.649, 'weight': 0.903, 'content': [{'end': 593.318, 'text': 'Remember, the rule here is that the rate of change of x is negative y, and the rate of change of y is x.', 'start': 585.649, 'duration': 7.669}, {'end': 601.447, 'text': 'Set up as vectors like this, we could rewrite the right-hand side of this equation as a product of this matrix with the original vector, xy.', 'start': 593.318, 'duration': 8.129}, {'end': 606.873, 'text': "The top row encodes Juliet's rule, and the bottom row encodes Romeo's rule.", 'start': 602.147, 'duration': 4.726}, {'end': 615.661, 'text': 'So what we have here is a differential equation telling us that the rate of change of some vector is equal to a certain matrix times itself.', 'start': 607.898, 'duration': 7.763}, {'end': 622.904, 'text': "In a moment, we'll talk about how matrix exponentiation solves this kind of equation.", 'start': 619.042, 'duration': 3.862}], 'summary': 'Differential equation relates rate of change to matrix multiplication.', 'duration': 37.255, 'max_score': 585.649, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo585649.jpg'}, {'end': 701.936, 'src': 'heatmap', 'start': 646.335, 'weight': 0.718, 'content': [{'end': 652.077, 'text': 'The basic idea is that when you multiply a matrix by the vector it pulls out the first column.', 'start': 646.335, 'duration': 5.742}, {'end': 658.399, 'text': 'And similarly, if you multiply it by that pulls out the second column.', 'start': 653.998, 'duration': 4.401}, {'end': 667.081, 'text': 'What this means is that when you look at a matrix, you can read its columns as telling you what it does to these two vectors,', 'start': 659.899, 'duration': 7.182}, {'end': 668.382, 'text': 'known as the basis vectors.', 'start': 667.081, 'duration': 1.301}, {'end': 676.517, 'text': "The way it acts on any other vector is a result of scaling and adding these two basis results by that vector's coordinates.", 'start': 669.335, 'duration': 7.182}, {'end': 686.58, 'text': 'So, looking back at the matrix from our system, notice how from its columns, we can tell it takes the first basis vector to and the second to.', 'start': 677.577, 'duration': 9.003}, {'end': 689.021, 'text': "hence why I'm calling it the 90-degree rotation matrix.", 'start': 686.58, 'duration': 2.441}, {'end': 697.033, 'text': "What it means for our equation is that it's saying wherever Romeo and Juliet are in this state space,", 'start': 691.209, 'duration': 5.824}, {'end': 701.936, 'text': 'their rate of change has to look like a 90 degree rotation of this position vector.', 'start': 697.033, 'duration': 4.903}], 'summary': 'Matrix multiplication represents 90-degree rotation in state space.', 'duration': 55.601, 'max_score': 646.335, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo646335.jpg'}, {'end': 701.936, 'src': 'embed', 'start': 677.577, 'weight': 3, 'content': [{'end': 686.58, 'text': 'So, looking back at the matrix from our system, notice how from its columns, we can tell it takes the first basis vector to and the second to.', 'start': 677.577, 'duration': 9.003}, {'end': 689.021, 'text': "hence why I'm calling it the 90-degree rotation matrix.", 'start': 686.58, 'duration': 2.441}, {'end': 697.033, 'text': "What it means for our equation is that it's saying wherever Romeo and Juliet are in this state space,", 'start': 691.209, 'duration': 5.824}, {'end': 701.936, 'text': 'their rate of change has to look like a 90 degree rotation of this position vector.', 'start': 697.033, 'duration': 4.903}], 'summary': 'Matrix depicts 90-degree rotation of position vector.', 'duration': 24.359, 'max_score': 677.577, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo677577.jpg'}, {'end': 768.608, 'src': 'heatmap', 'start': 717.236, 'weight': 4, 'content': [{'end': 725.501, 'text': 'More specifically, since the length of this velocity vector equals the length of the position vector, then, for each unit of time,', 'start': 717.236, 'duration': 8.265}, {'end': 730.705, 'text': "the distance that this covers is equal to one radius's worth of arc length along that circle.", 'start': 725.501, 'duration': 5.204}, {'end': 735.468, 'text': 'In other words, it rotates at one radian per unit time.', 'start': 732.426, 'duration': 3.042}, {'end': 740.591, 'text': 'So in particular, it would take two pi units of time to make a full revolution.', 'start': 736.148, 'duration': 4.443}, {'end': 749.392, 'text': 'If you want to describe this kind of rotation with a formula, we can use a more general rotation matrix, which looks like this.', 'start': 742.987, 'duration': 6.405}, {'end': 752.235, 'text': 'Again, we can read it in terms of the columns.', 'start': 750.493, 'duration': 1.742}, {'end': 753.115, 'text': 'Notice how.', 'start': 752.755, 'duration': 0.36}, {'end': 760.081, 'text': 'the first column tells us that it takes that first basis vector to cosine of t, sine of t,', 'start': 753.115, 'duration': 6.966}, {'end': 765.725, 'text': 'and the second column tells us that it takes the second basis vector to negative sine of t, cosine of t,', 'start': 760.081, 'duration': 5.644}, {'end': 768.608, 'text': 'both of which are consistent with rotating by t radians.', 'start': 765.725, 'duration': 2.883}], 'summary': 'Velocity vector equals position vector, rotates at one radian per unit time, takes two pi units for full revolution.', 'duration': 43.107, 'max_score': 717.236, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo717236.jpg'}], 'start': 394.507, 'title': 'Quantum mechanics and matrix-based differential equations', 'summary': 'Discusses differential equations modeling the dynamic love relationship between romeo and juliet, and showcases the geometric interpretation of matrix-based differential equations, including the 90-degree rotation matrix and its relation to circular motion.', 'chapters': [{'end': 593.318, 'start': 394.507, 'title': 'Quantum mechanics and relationships', 'summary': 'Discusses a system of differential equations modeling the dynamic love relationship between romeo and juliet, and the use of matrix exponents to systematically solve more general versions of the equation without guessing and checking.', 'duration': 198.811, 'highlights': ["The system of differential equations models the dynamic love relationship between Romeo and Juliet, where the rate of change of Juliet's love is equal to negative one times Romeo's love for her, and the rate of change of Romeo's love is equal to the size of Juliet's love.", 'The goal is to systematically solve more general versions of the equation without guessing and checking, and to find explicit functions for x and y that match the initial set of conditions at time t equals zero.', "The relationship between Romeo and Juliet is represented as a point in a 2D space, with the x-coordinate capturing Juliet's feelings and the y-coordinate capturing Romeo's, and it's helpful to picture this state as an arrow from the origin."]}, {'end': 768.608, 'start': 593.318, 'title': 'Matrix-based differential equations and geometric interpretation', 'summary': "Explores the geometric interpretation of matrix-based differential equations, showcasing the 90-degree rotation matrix and its relation to circular motion, where the velocity vector rotates at one radian per unit time, taking two pi units of time to make a full revolution, thus exemplifying the matrix's role in visualizing and solving such equations.", 'duration': 175.29, 'highlights': ['The matrix from our system is a 90-degree rotation matrix.', 'The rate of change has no component in the direction of the position, implying circular motion, rotating at one radian per unit time and taking two pi units of time to make a full revolution.', 'The general rotation matrix, used to describe this kind of rotation, takes the first basis vector to cosine of t, sine of t, and the second basis vector to negative sine of t, cosine of t.']}], 'duration': 374.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo394507.jpg', 'highlights': ["The system of differential equations models the dynamic love relationship between Romeo and Juliet, where the rate of change of Juliet's love is equal to negative one times Romeo's love for her, and the rate of change of Romeo's love is equal to the size of Juliet's love.", "The relationship between Romeo and Juliet is represented as a point in a 2D space, with the x-coordinate capturing Juliet's feelings and the y-coordinate capturing Romeo's, and it's helpful to picture this state as an arrow from the origin.", 'The goal is to systematically solve more general versions of the equation without guessing and checking, and to find explicit functions for x and y that match the initial set of conditions at time t equals zero.', 'The matrix from our system is a 90-degree rotation matrix.', 'The general rotation matrix, used to describe this kind of rotation, takes the first basis vector to cosine of t, sine of t, and the second basis vector to negative sine of t, cosine of t.', 'The rate of change has no component in the direction of the position, implying circular motion, rotating at one radian per unit time and taking two pi units of time to make a full revolution.']}, {'end': 1166.694, 'segs': [{'end': 804.888, 'src': 'heatmap', 'start': 780.409, 'weight': 0.771, 'content': [{'end': 791.257, 'text': 'The active viewers among you might also enjoy taking a moment to pause and confirm that the explicit formulas you get out of this for x of t and y of t really do satisfy the system of differential equations that we started with.', 'start': 780.409, 'duration': 10.848}, {'end': 804.888, 'text': "The mathematician in, you might wonder if it's possible to solve not just this specific system, but equations like it for any other matrix,", 'start': 798.103, 'duration': 6.785}], 'summary': 'Viewers can confirm if explicit formulas satisfy differential equations.', 'duration': 24.479, 'max_score': 780.409, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo780409.jpg'}, {'end': 845.278, 'src': 'embed', 'start': 820.06, 'weight': 1, 'content': [{'end': 825.425, 'text': 'And on the flip side, how being able to compute matrix exponents lets you explicitly solve this equation.', 'start': 820.06, 'duration': 5.365}, {'end': 829.922, 'text': "A much less whimsical example is Schrödinger's famous equation,", 'start': 826.539, 'duration': 3.383}, {'end': 834.887, 'text': 'which is the fundamental equation describing how systems in quantum mechanics change over time.', 'start': 829.922, 'duration': 4.965}, {'end': 838.531, 'text': "It looks pretty intimidating, and I mean it's quantum mechanics.", 'start': 835.688, 'duration': 2.843}, {'end': 842.294, 'text': "so of course it will, but it's actually not that different from the Romeo-Juliet setup.", 'start': 838.531, 'duration': 3.763}, {'end': 845.278, 'text': 'This symbol here refers to a certain vector.', 'start': 843.035, 'duration': 2.243}], 'summary': "Computing matrix exponents can explicitly solve equations, like schrödinger's famous equation in quantum mechanics.", 'duration': 25.218, 'max_score': 820.06, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo820060.jpg'}, {'end': 890.462, 'src': 'heatmap', 'start': 857.871, 'weight': 0.946, 'content': [{'end': 863.457, 'text': 'The equation says that the rate at which this state vector changes looks like a certain matrix times itself.', 'start': 857.871, 'duration': 5.586}, {'end': 869.728, 'text': "There are a number of things that make Schrodinger's equation notably more complicated, but in the back of your mind,", 'start': 864.625, 'duration': 5.103}, {'end': 875.732, 'text': 'you might think of it as a target point that you and I can build up to with simpler examples like Romeo and Juliet,', 'start': 869.728, 'duration': 6.004}, {'end': 878.014, 'text': 'offering more friendly stepping stones along the way.', 'start': 875.732, 'duration': 2.282}, {'end': 884.938, 'text': 'Actually, the simplest example, which is tied to ordinary real-number powers of e, is the one-dimensional case.', 'start': 879.575, 'duration': 5.363}, {'end': 890.462, 'text': 'This is when you have a single changing value, and its rate of change equals some constant times itself.', 'start': 885.379, 'duration': 5.083}], 'summary': "Schrodinger's equation represents change as matrix multiplication with simpler examples like one-dimensional case.", 'duration': 32.591, 'max_score': 857.871, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo857871.jpg'}, {'end': 1033.826, 'src': 'heatmap', 'start': 1006.499, 'weight': 0.797, 'content': [{'end': 1014.201, 'text': 'You see, up in the two-dimensional case, when we have a changing vector whose rate of change is constrained to be some matrix times itself.', 'start': 1006.499, 'duration': 7.702}, {'end': 1019.262, 'text': 'what the solution looks like is also an exponential term, acting on a given initial condition.', 'start': 1014.201, 'duration': 5.061}, {'end': 1026.124, 'text': 'But the exponential part, in that case, will produce a matrix that changes with time, and the initial condition is a vector.', 'start': 1019.882, 'duration': 6.242}, {'end': 1033.826, 'text': 'In fact, you should think of the definition of matrix exponentiation as being heavily motivated by making sure that this fact is true.', 'start': 1026.864, 'duration': 6.962}], 'summary': 'Matrix exponentiation produces a changing matrix with time, motivated to ensure exponential term acts on initial condition.', 'duration': 27.327, 'max_score': 1006.499, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1006499.jpg'}, {'end': 1067.024, 'src': 'embed', 'start': 1026.864, 'weight': 0, 'content': [{'end': 1033.826, 'text': 'In fact, you should think of the definition of matrix exponentiation as being heavily motivated by making sure that this fact is true.', 'start': 1026.864, 'duration': 6.962}, {'end': 1043.503, 'text': 'For example, if we look back at the system that popped up with Romeo and Juliet, the claim now is that solutions look like e raised to this 0,', 'start': 1034.997, 'duration': 8.506}, {'end': 1048.768, 'text': 'negative, 1,, 1, 0 matrix, all times time multiplied by some initial condition.', 'start': 1043.503, 'duration': 5.265}, {'end': 1051.491, 'text': "But we've already seen the solution in this case.", 'start': 1049.649, 'duration': 1.842}, {'end': 1054.573, 'text': 'We know it looks like a rotation matrix times the initial condition.', 'start': 1051.671, 'duration': 2.902}, {'end': 1062.219, 'text': "So let's take a moment to roll up our sleeves and compute the exponential term using the definition that I mentioned at the start and see if it lines up.", 'start': 1055.274, 'duration': 6.945}, {'end': 1067.024, 'text': 'Remember, writing e to the power of a matrix is a shorthand.', 'start': 1063.24, 'duration': 3.784}], 'summary': 'Matrix exponentiation is defined using e to the power of a matrix and its solutions align with rotation matrices.', 'duration': 40.16, 'max_score': 1026.864, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1026864.jpg'}], 'start': 769.781, 'title': 'Matrix exponential applications', 'summary': "Explores matrix exponentiation's use in predicting outcomes and solving differential equations, with applications in various fields such as finance, population growth, and epidemic modeling, emphasizing the connection between exponential growth and matrix exponentiation.", 'chapters': [{'end': 856.93, 'start': 769.781, 'title': 'Matrix exponents and differential equations', 'summary': "Discusses the use of matrix multiplication to predict outcomes in a system, the concept of matrix exponents for solving differential equations, and the analogy between solving the romeo-juliet system and schrödinger's equation in quantum mechanics.", 'duration': 87.149, 'highlights': ['The main goal is to understand how matrix multiplication and exponentiation intuitively depict operations and explicitly solve equations.', 'The chapter discusses using matrix multiplication to predict outcomes in a system after a certain duration of time.', "It explores the concept of matrix exponents for solving differential equations and highlights the analogy between solving the Romeo-Juliet system and Schrödinger's equation in quantum mechanics.", 'The chapter also mentions confirming that the explicit formulas for x of t and y of t satisfy the system of differential equations.']}, {'end': 1166.694, 'start': 857.871, 'title': 'Exponential growth and matrix exponents', 'summary': "Discusses the schrodinger's equation and its relation to exponential growth, including its applications in finance, population growth, and epidemic modeling, as well as the concept of matrix exponentiation and its application in solving systems of differential equations, demonstrating the connection between exponential growth and matrix exponentiation.", 'duration': 308.823, 'highlights': ["Schrodinger's equation relates to exponential growth and has applications in finance, population growth, and epidemic modeling, providing solutions for different initial conditions.", 'The concept of matrix exponentiation is motivated by the solution of systems of differential equations, demonstrating the connection between exponential growth and matrix exponentiation.', 'The computation of the exponential term using the Taylor series for e to the x results in a cycling pattern, yielding the rotation matrix and showcasing the beauty of reasoning about the same system in different ways.']}], 'duration': 396.913, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo769781.jpg', 'highlights': ['Matrix exponentiation used to predict outcomes in a system after a certain duration of time', "Schrodinger's equation relates to exponential growth and has applications in finance, population growth, and epidemic modeling", "Matrix exponents used for solving differential equations, highlighting the analogy between solving the Romeo-Juliet system and Schrödinger's equation", 'Exponential term computed using the Taylor series for e to the x results in a cycling pattern, yielding the rotation matrix']}, {'end': 1601.421, 'segs': [{'end': 1193.385, 'src': 'embed', 'start': 1167.694, 'weight': 0, 'content': [{'end': 1174.139, 'text': 'Hopefully, the fact that these line up inspires a little confidence in the claim that matrix exponents really do solve systems like this.', 'start': 1167.694, 'duration': 6.445}, {'end': 1181.525, 'text': 'This explains the computation we saw at the start, by the way, with the matrix that had negative pi and pi off the diagonals,', 'start': 1175.32, 'duration': 6.205}, {'end': 1182.746, 'text': 'producing the negative identity.', 'start': 1181.525, 'duration': 1.221}, {'end': 1188.621, 'text': 'This expression is exponentiating a 90-degree rotation matrix times pi,', 'start': 1183.536, 'duration': 5.085}, {'end': 1193.385, 'text': 'which is another way to describe what the Romeo-Juliet setup does after pi units of time.', 'start': 1188.621, 'duration': 4.764}], 'summary': 'Matrix exponents solve systems, as seen with 90-degree rotation matrix times pi.', 'duration': 25.691, 'max_score': 1167.694, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1167694.jpg'}, {'end': 1289.578, 'src': 'embed', 'start': 1248.775, 'weight': 2, 'content': [{'end': 1267.081, 'text': "In all of these cases we have this really neat general idea that if you take some operation that rotates 90 degrees in some plane often it's a plane in some high-dimensional space that we can't visualize then what we get by exponentiating that operation times time is something that generates all other rotations in that same plane.", 'start': 1248.775, 'duration': 18.306}, {'end': 1273.245, 'text': "One of the more complicated variations on this same theme is Schrodinger's equation.", 'start': 1269.542, 'duration': 3.703}, {'end': 1278.729, 'text': "It's not just that this has the derivative of a state equals some matrix times that state form.", 'start': 1273.765, 'duration': 4.964}, {'end': 1284.734, 'text': 'The nature of the relevant matrix here is such that the equation also describes a kind of rotation.', 'start': 1279.45, 'duration': 5.284}, {'end': 1289.578, 'text': "Though in many applications of Schrodinger's equation, it'll be a rotation in a kind of function space.", 'start': 1285.234, 'duration': 4.344}], 'summary': "Exponentiating a 90-degree rotation operation generates all other rotations in a high-dimensional space, exemplified in schrodinger's equation.", 'duration': 40.803, 'max_score': 1248.775, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1248775.jpg'}, {'end': 1361.185, 'src': 'embed', 'start': 1331.182, 'weight': 1, 'content': [{'end': 1334.464, 'text': 'But matrix exponentiation can do so much more than just rotation.', 'start': 1331.182, 'duration': 3.282}, {'end': 1338.927, 'text': 'You can always visualize these sorts of differential equations using a vector field.', 'start': 1335.085, 'duration': 3.842}, {'end': 1345.778, 'text': 'The idea is that this equation tells us the velocity of a state is entirely determined by its position.', 'start': 1340.595, 'duration': 5.183}, {'end': 1354.422, 'text': 'So what we do is go to every point in the space and draw a little vector indicating what the velocity of a state must be if it passes through that point.', 'start': 1346.418, 'duration': 8.004}, {'end': 1361.185, 'text': 'For our type of equation, this means that we go to each point v in space and we attach the vector m times v.', 'start': 1355.322, 'duration': 5.863}], 'summary': 'Matrix exponentiation enables visualization of differential equations using vector fields, determining state velocity by position.', 'duration': 30.003, 'max_score': 1331.182, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1331182.jpg'}, {'end': 1601.421, 'src': 'embed', 'start': 1563.739, 'weight': 3, 'content': [{'end': 1570.323, 'text': 'So the derivative of the expression is m times the original expression, and hence it solves the equation.', 'start': 1563.739, 'duration': 6.584}, {'end': 1574.806, 'text': 'This actually sweeps under the rug some details required for rigor,', 'start': 1571.403, 'duration': 3.403}, {'end': 1579.729, 'text': 'mostly centered around the question of whether or not this thing actually converges, but it does give the main idea.', 'start': 1574.806, 'duration': 4.923}, {'end': 1585.274, 'text': 'In the next chapter, I would like to talk more about the properties that this operation has,', 'start': 1581.072, 'duration': 4.202}, {'end': 1588.675, 'text': 'most notably its relationship with eigenvectors and eigenvalues,', 'start': 1585.274, 'duration': 3.401}, {'end': 1594.758, 'text': 'which leads us to more concrete ways of thinking about how you actually carry out this computation, which otherwise seems insane.', 'start': 1588.675, 'duration': 6.083}, {'end': 1601.421, 'text': 'Also, time permitting, it might be fun to talk about what it means to raise e to the power of the derivative operator.', 'start': 1596.059, 'duration': 5.362}], 'summary': 'The derivative of the expression is m times the original, leading to discussions on convergence, properties, eigenvectors, eigenvalues, and computation methods.', 'duration': 37.682, 'max_score': 1563.739, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1563739.jpg'}], 'start': 1167.694, 'title': 'Matrix exponents in differential equations', 'summary': "Explores the application of matrix exponents in solving systems and visualizing state evolution in differential equations, including rotation in state space, negative identity, and its relationship with the derivative operator, with implications in math and physics, such as schrodinger's equation.", 'chapters': [{'end': 1304.749, 'start': 1167.694, 'title': 'Matrix exponents and rotation', 'summary': "Explains how matrix exponents solve systems by describing rotation, such as a 90-degree rotation in state space, leading to a negative identity and analogous to imaginary number exponents, with applications in math and physics, including schrodinger's equation.", 'duration': 137.055, 'highlights': ['Matrix exponents solve systems by describing rotation, such as a 90-degree rotation in state space, leading to a negative identity.', "Applications in math and physics, including Schrodinger's equation, involve rotation and complex numbers.", 'Exponentiating some object which acts as a 90 degree rotation times time generates all other rotations in the same plane.']}, {'end': 1601.421, 'start': 1304.749, 'title': 'Matrix exponentiation in differential equations', 'summary': 'Explains how matrix exponentiation visualizes the evolution of states in differential equations using vector fields, demonstrating the transformation from initial conditions to time t and its relationship with the derivative operator.', 'duration': 296.672, 'highlights': ['Matrix exponentiation visualizes the evolution of states in differential equations using vector fields, allowing visualization of the transition from initial conditions to time t, as demonstrated with the 90-degree rotation matrix and the Romeo and Juliet example.', 'The derivative of the expression for e to the mt is m times the original expression, showing its relationship with the derivative operator and its ability to solve the equation.', 'The chapter highlights the properties and relationship of matrix exponentiation with eigenvectors and eigenvalues, offering concrete ways of carrying out the computation and discusses raising e to the power of the derivative operator.']}], 'duration': 433.727, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/O85OWBJ2ayo/pics/O85OWBJ2ayo1167694.jpg', 'highlights': ['Matrix exponents solve systems by describing rotation, such as a 90-degree rotation in state space, leading to a negative identity.', 'Matrix exponentiation visualizes the evolution of states in differential equations using vector fields, allowing visualization of the transition from initial conditions to time t, as demonstrated with the 90-degree rotation matrix and the Romeo and Juliet example.', 'Exponentiating some object which acts as a 90 degree rotation times time generates all other rotations in the same plane.', 'The derivative of the expression for e to the mt is m times the original expression, showing its relationship with the derivative operator and its ability to solve the equation.', "Applications in math and physics, including Schrodinger's equation, involve rotation and complex numbers.", 'The chapter highlights the properties and relationship of matrix exponentiation with eigenvectors and eigenvalues, offering concrete ways of carrying out the computation and discusses raising e to the power of the derivative operator.']}], 'highlights': ['Matrix exponents used to solve differential equations in quantum mechanics', 'Exponential of a matrix defined using Taylor series, approaching e times e', 'Matrix exponents visualize evolution of states in differential equations', 'Matrix exponentiation predicts outcomes in finance and epidemic modeling', 'Matrix exponents describe rotation in state space, including 90-degree rotation', 'Matrix exponentiation involves multiplying the matrix by itself to obtain powers', 'System of differential equations models dynamic love relationship between Romeo and Juliet', "Schrodinger's equation relates to exponential growth and has applications in finance", "Matrix exponentiation highlights the analogy between solving Romeo-Juliet system and Schrödinger's equation", 'Exponentiating some object which acts as a 90 degree rotation times time generates all other rotations in the same plane']}