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https://github.com/particle1331/computational-linear-algebra
Rapidly develops the SVD and uses it for everything.
https://github.com/particle1331/computational-linear-algebra
a linear-algebra loss-surface math mathematics matrix moore-penrose-pseudoinverse proof svd
Last synced: 5 days ago
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Rapidly develops the SVD and uses it for everything.
- Host: GitHub
- URL: https://github.com/particle1331/computational-linear-algebra
- Owner: particle1331
- Created: 2021-01-22T07:58:11.000Z (almost 4 years ago)
- Default Branch: master
- Last Pushed: 2023-01-25T13:25:46.000Z (almost 2 years ago)
- Last Synced: 2024-12-02T19:09:02.372Z (about 2 months ago)
- Topics: a, linear-algebra, loss-surface, math, mathematics, matrix, moore-penrose-pseudoinverse, proof, svd
- Language: Jupyter Notebook
- Homepage:
- Size: 44.9 MB
- Stars: 4
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Computational Linear Algebra
Notes and code experiments for linear algebra in code. The idea is to construct the SVD as soon as possible, then use it for everything else — from characterizing invertibility, to parametrizing the loss surface of a linear regression model. Some of the interesting stuff that are covered:
* Proof of the real spectral theorem, and a code demo
* Proof of the singular value decomposition (SVD)
* An extensive discussion of the Moore-Penrose pseudoinverse
* Stability of the Gram-Schmidt algorithm
* Characterizing the loss surface of a linear regression problem
* Characterizing quadratic forms using the principal axes theorem
Figure. SVD of a sum of Gaussians. Only the first few vectors are meaningful, the rest model noise.
Figure. Energy surface of an indefinite matrix. It has a negative minimum and a positive maximum.
## Contents
1. [Vectors and matrices](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/01-vectors.ipynb)
2. [Singular value decomposition](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/02-svd.ipynb)
3. [Matrix multiplication and norms](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/03-norms.ipynb)
4. [Rank and dimension](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/04-rank.ipynb)
5. [Four fundamental subspaces](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/05-four-subspaces.ipynb)
6. [Determinant](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/06-det.ipynb)
7. [Matrix inverse and pseudoinverse](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/07-inverse.ipynb)
8. [Projection and orthogonalization](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/08-projection.ipynb)
9. [Least squares for model fitting](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/09-least-squares.ipynb)
10. [Eigendecomposition](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/10-eigendecomp.ipynb)
11. [Quadratic form and definiteness](https://github.com/particle1331/computational-linear-algebra/blob/master/chapters/11-quadratic.ipynb)
## Quick links
* [Proofs involving the Moore-Penrose pseudoinverse](https://en.wikipedia.org/wiki/Proofs_involving_the_Moore%E2%80%93Penrose_inverse)
* [KaTeX Supported Functions](https://katex.org/docs/supported.html)
## References
* [Mike X Cohen.](http://mikexcohen.com/) [*Complete linear algebra: theory and implementation in code*. Udemy. (2021)](https://www.udemy.com/course/linear-algebra-theory-and-implementation/)
* [Sheldon Axler. *Down With Determinants!* The American Monthly. (1996)](https://www.maa.org/sites/default/files/pdf/awards/Axler-Ford-1996.pdf)
* [Leslie Hogben (editor). *Handbook of Linear Algebra*. CRC Press. (2014)](https://www.oreilly.com/library/view/handbook-of-linear/9781466507296/)
* [Cleve Moler. *Numerical Computing with MATLAB*. The MathWorks / SIAM. (2013)](https://www.mathworks.com/moler/index_ncm.html)
* [Peter Olver and Chehzrad Shakiban. *Applied Linear Algebra*. UTM Springer. (2018)](https://www-users.math.umn.edu/~olver/books.html)
* [Petersen & Pedersen. *The Matrix Cookbook*. v. Nov. 15, 2012. (2012)](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf)