#vectors #linearalgebra #matrices #eigenvectors #eigenvalues

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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.

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These are the resources I mentioned in the video:

[MTB 1]    • Linear Algebra 16c1: The Sum is the T...  
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]    • Linear Algebra 15c: The Reflection Tr...  
Reflections and their eigenvalues.

[MTB 3]    • Linear Algebra 15d: The Projection Tr...  
Projections and their eigenvalues.

[MTB 4]    • Linear Algebra 15h: The Derivative as...  
The derivative as a linear transformation, including its eigenfunctions.

[MTB 5]    • Linear Algebra 16k: Eigenvalues, Eige...  
The matrix similarity transformation. Similar matrices have the same eigenvalues, and related eigenvectors.

[MTB 6]    • Linear Algebra 16n: Every Matrix Sati...  
Every matrix satisfies its own characteristic polynomial. This is pretty advanced, but still an amazing property of matrices.

[3B1B 1]    • Eigenvectors and eigenvalues | Chapte...  
An introduction to eigenvalues and eigenvectors.

[ZS 1]    • The Applications of Matrices | What I...  
[ZS 2]    • The applications of eigenvectors and ...  
Applications of the eigenvalue decomposition in practice, including Fibonacci numbers, clustering, a mass oscillating on a spring, and even a zombie apocalypse.

[TB 1]    • Visualizing Diagonalization & Eigenbases  
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

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