QuantEcon / QuantEcon/lecture-python.myst
svd_intro: add the Courant-Fischer min-max characterisation as a route to PCA
Nobody has claimed this yet.
- Dominant language
- TeX
- Stars
- 123
- Forks
- 57
- Avg merge
- 3d 10h
- Merged PRs (30d)
- 11
Description
svd_intro.md currently derives principal components three ways — as an application of the SVD (Application: Principal Components Analysis (PCA)), through the explicit Relationship of PCA to SVD, and again in PCA with Eigenvalues and Eigenvectors. What none of them gives is the variational characterisation: that the principal components are the successive maximisers of the Rayleigh quotient, which is what the Courant–Fischer min-max theorem states.
That framing answers a question the current derivations leave open — not just what the principal components are, but why they are optimal, and in what precise sense the k-th component is the best remaining direction.
This was requested in QuantEcon/meta#28, Tom Sargent's brief on eigenvalues and eigenfunctions, which asked to "use principal components analysis as an application of the min-max theorem". That issue has been closed against the material that did ship; this is one of the residuals it did not cover, spun out so it is not lost.
Suggested shape: a section after PCA with Eigenvalues and Eigenvectors stating the Rayleigh quotient, giving the Courant–Fischer characterisation for a symmetric matrix, and connecting it back to the components already computed earlier in the lecture — ideally reusing the same worked example so the reader sees the two routes agree.
The original brief also links the Wikipedia min-max theorem article as a starting reference.
Contributor guide
No contributing guide indexed for this repository
First steps
- Read the whole issue, then the project's contributing guide.
- Comment on the issue to say you are picking it up — it saves two people doing the same work.
- Fork the repository and make your change on a branch.
- Open a pull request that references the issue number.
Research direction
Read svd_intro.md, especially “PCA with Eigenvalues and Eigenvectors” and the earlier PCA derivations, then consult the linked min-max theorem reference. Add a section stating the Rayleigh quotient and Courant–Fischer characterisation for symmetric matrices, reusing the worked example to connect both routes and show the successive optimal directions.
Written by the indexing model from the issue text.
Assessment
- Tech stack
- markdown
- Domain
- documentation
- Issue type
- Documentation
- Difficulty
- 4/5
- Estimated time
- 3-5 days
- Activity status
- Quiet
- Clarity
- Mostly clear
- Newbie friendliness
- 56/100