Further Matrix Decompositions: LU, Cholesky, QR, and SVD

We've learned about matrix diagonalization, which is a type of matrix decomposition. There are some others that we must learn, including LU decomposition, Cholesky decomposition, QR decomposition, and singular value decomposition. Let's get a closer look!

Script by Lorcan Nicholls

Check out "Is This Wi-Fi Organic?", my book on disarming pseudoscience!

[cgifrog]

Would love to see more math videos. Maybe differential equations, graph theory, discrete mathematics, or even some analysis would be amazing but perhaps out of scope. Your Linear Algebra and Calculus videos carried me last semester, thanks for all that you do!

[Hyperviolet]

FINALLY we're getting more maths videos!!!

[tristancools]

I was yesterday just rewatching the whole linear algebra series, its like a gift to get this new linear algebra video

[MOSMASTERING]

One of my favourite quotes about science actually came from Ricky Gervais..

He said - "If all science was wiped out, it would still be true, and someone would rediscover it. If you wiped out religion, it would never come back in the same form.""

Facts are facts, truth is truth - and as we slowly learn more, we get closer to it. The laws, constants, rules and fundamental constituents of reality will always be the same.

In addition to physics, biology would be rediscovered from scratch to be the same as well. Nobody would stop at intelligent design. As soon as we discovered DNA it was game over for ID, all the clues are there written into it.

[FishSticker]

Let's see the flat earthers debunk this one

[pwd-tk]

This video is underrated. Most of this stuff is relevant to learning ML stuff.

[GuapoTaco]

Just in time for winter break now that I’m done with linear algebra

[Joel-The-Bad]

Dave did you see the final experiment Antarctica flat earth destruction video? I know you were talking about it at one point. It happened. Lmao

[backyard]

I'm a math undergraduate and we didn't learn this in Linear Algebra. Thanks dave

[void]

thanks professor dave, this will help me find eigenvectors for my wave conjugation homework

[Blingsss]

Amazing video! Matrix decompositions are fascinating tools for simplifying complex linear algebra problems. I’ve always found LU and QR decomposition particularly helpful in solving systems of equations, while SVD opened my eyes to its importance in data science and PCA. SolutionInn has been a lifesaver for breaking down these topics with clear examples and resources. It’s amazing how these techniques connect to so many applications from optimizing algorithms to understanding transformations. Definitely worth diving deeper into.

[undine]

If, for some matrix A, you compute A = QR using QR decomposition, and then set A to RQ, and repeat many many times, legend has it that eventually A has its original eigenvalues listed on its diagonal.

[kevwiththebev]

You have a gift for teaching, have you considered making a series about computer science?

[lksr]

Thank you so much dave! I was waiting for more math videos! You are the best!!

[carrizzle]

I took Finite Mathematics (Matrix algebra and probability) before I joined the engineering program and then took Linear Algebra and I didn't know that algebra could get that difficult. A lot of rules that relate to vectors in space.🤯

[sndrrz]

Mathematics returns! Bitterly ironic for me as I was hoping to take linalg next semester but it wasn't offered and I'll have to wait until next fall :(.

[TwoCraZyEyes]

Dave when are you doing a video about the final experiment?

[ticonderogafutterman]

Also Jordan decomposition and Schur decomposition might be mentioned here... (if possible)

[pmsoltan]

For the Cholesky decomposition you need to assume that A is not only hermitian, but also positive (some people call this "positive definite"). Also the introduction to the SVD is way off - the problem is not in A having "repeated eigenvalues" (you mean its eigenvalues have non-trivial multiplicities), but the decomposition itself is presented correctly.

[jessicatymczak]

So, how would you get the density matrix, sum^n_i phi_i * phi_i n<N, using only matrix multiplication, matrix addition, and scalar multiplication.