Eigenvalues and eigenvectors | Linear algebra episode 8

#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: 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: Th...
Reflections and their eigenvalues.
[MTB 3] • Linear Algebra 15d: Th...
Projections and their eigenvalues.
[MTB 4] • Linear Algebra 15h: Th...
The derivative as a linear transformation, including its eigenfunctions.
[MTB 5] • Linear Algebra 16k: Ei...
The matrix similarity transformation. Similar matrices have the same eigenvalues, and related eigenvectors.
[MTB 6] • Linear Algebra 16n: Ev...
Every matrix satisfies its own characteristic polynomial. This is pretty advanced, but still an amazing property of matrices.
[3B1B 1] • Eigenvectors and eigen...
An introduction to eigenvalues and eigenvectors.
[ZS 1] • The Applications of Ma...
[ZS 2] • The applications of ei...
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 Diagonaliz...
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
( creativecommons.org/licenses/... )

[@jneal4154]

3:40 Ummm... I thought that diagonal matrices could be rectangular? Isn't singular value decomposition usually preferred over eigenvalue decomposition because the D in UDV^T can be rectangular?
Sure, you can't conveniently take powers of D like with eigenvalue decomposition, but it is still a diagonal matrix is it not?

[@AllAnglesMath]

That's an excellent point. In fact, in the upcoming video about the SVD, we do indeed show a diagonal rectangular matrix, and we explicitly call it diagonal. My bad. I should have been more clear; the point was that in this specific video, we're only dealing with square matrices. Thanks for the correction!

[@Grateful92]

I thank UA-cam for bringing me this beautiful and thoughtful video. This video has given me a new axiom to consider while playing with the power set of existence. I love linear algebra and its because of people like you who really know how to teach maths, thanks for existing and uploading this video on UA-cam.

[@AllAnglesMath]

My parents are responsible for the fact that I exist, so I will pass on the message 😉

[@fedebonons8453]

35 minutes already know this gonna be a banger

[@AllAnglesMath]

There was no way to do it in less time. Thanks for having such confidence in our videos 🤟

[@fedebonons8453]

@@AllAnglesMath yeah of course when its about this channel the longer the better :)

[@johnstuder847]

Thank you for bringing Linear Algebra to life. It’s amazing to me how such a powerful, practical, and in many ways simple subject can be obscured by mathematicians taking the strictly abstract approach. It’s sad that many scientists and engineers first exposure to Linear Algebra is in a math class. Most leave the class with no idea what an eigenvalue is, and without an intro to the SVD. I think that is criminal. Thank you again for an excellent video and series!

[@AllAnglesMath]

Thank you so much!

[@tantzer6113]

Pedagogically, this may be the best video on the subject!

[@AllAnglesMath]

Thank you!

[@datamoon]

good stuff man

[@MusicEngineeer]

When thinking about rotation matrices, one observes that points on concentric circles centered at the origin will always remain on the same circle. I wonder, if it would be meaningful to call these circles "eigencurves" for this reason. Could the idea of such "eigencurves" be a meaningful generalization of the idea of eigenvectors and "eigenlines"? I guess, a general 2x2 matrix would have some sort of conic section as "eigencurve"? I know that in dynamical systems theory there is this idea of invariant sets and I think that in a 2D linear system, these invariant sets would be precisely those conic sections?