Eigenvalues and eigenvectors | Linear algebra episode 8

#vectors #linearalgebra #matrices #eigenvectors #eigenvalues Exclusive videos on Patreon: https://www.patreon.com/user?u=86649007 What is an eigenvector? How can we turn an arbitrary matrix into a diagonal one? How can we use this to study the long-term behavior of an ecosystem? In this video, you will learn about diagonals, decoupling, and the eating habits of unicorns. To help us make more content, and to get access to new videos many months before they appear on Youtube, consider supporting us on Patreon: https://www.patreon.com/user?u=86649007 These are the resources I mentioned in the video: [MTB 1] https://www.youtube.com/watch?v=OLl_reBXY-g&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=21 Explains why the sum of the eigenvalues is the trace, and their product is the determinant. This is an extremely beautiful and intuitive line of reasoning, based on the algebra of polynomials. [MTB 2] https://www.youtube.com/watch?v=y2WqIXrjyC4&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=2 Reflections and their eigenvalues. [MTB 3] https://www.youtube.com/watch?v=qxxo-a9snhw&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=3 Projections and their eigenvalues. [MTB 4] https://www.youtube.com/watch?v=lIwGFIFRspw&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=7 The derivative as a linear transformation, including its eigenfunctions. [MTB 5] https://www.youtube.com/watch?v=xk6W9vZF80Q&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=36 The matrix similarity transformation. Similar matrices have the same eigenvalues, and related eigenvectors. [MTB 6] https://www.youtube.com/watch?v=2pne2dzlNE4&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=39 Every matrix satisfies its own characteristic polynomial. This is pretty advanced, but still an amazing property of matrices. [3B1B 1] https://www.youtube.com/watch?v=PFDu9oVAE-g An introduction to eigenvalues and eigenvectors. [ZS 1] https://www.youtube.com/watch?v=rowWM-MijXU&list=PLi5WqFHu_OJNN-2rHmEOy7NZRNQ43qOuq&index=13 [ZS 2] https://www.youtube.com/watch?v=i8FukKfMKCI Applications of the eigenvalue decomposition in practice, including Fibonacci numbers, clustering, a mass oscillating on a spring, and even a zombie apocalypse. [TB 1] https://www.youtube.com/watch?v=EJG6gBeVdfw&list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6&index=70 Good visualization of what the change to an eigenbasis looks like on a grid. 0:00 The effect of a matrix on a circle 2:14 Diagonal matrices are fully decoupled 3:46 Finding the eigenvectors visually 6:38 Finding the eigenvectors using algebra 7:56 Trace and determinant 9:17 Not all matrices have eigenvectors 10:53 More examples and a few surprises 14:04 Eigenlines always go through the origin 14:50 The eigenvalues of a projection 17:17 An eigenvector for all permutation matrices 18:09 The eigenfunctions of the derivative operator 19:07 How to diagonalize a matrix 21:52 Similar matrices 23:07 Unicorns and trolls: population dynamics 27:09 Long-term stability of a system 29:15 More details 33:34 Mandelbrot animation This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )