📉PCA: Dimensionality Reduction from Eigenvectors
Connect PCA to the eigenvectors of the covariance matrix, then compress a 50-feature dataset to 5 components and defend exactly how much information you kept.
Phase 1Why Principal Components Are Eigenvectors of the Covariance Matrix
See why principal components are covariance eigenvectors
PCA isn't compression — it's rotation
6 minPCA isn't compression — it's rotation
The covariance matrix is a map of how your features move together
6 minThe covariance matrix is a map of how your features move together
Eigenvectors of the covariance matrix point along maximum variance
7 minEigenvectors of the covariance matrix point along maximum variance
Scale your features before PCA, or the largest number wins
5 minScale your features before PCA, or the largest number wins
Phase 2Running PCA on Real Data and Picking the Right Number of Components
Run PCA, plot variance, pick component count
PCA in five lines of sklearn — and why each line matters
6 minPCA in five lines of sklearn — and why each line matters
The scree plot tells you exactly how many components to keep
6 minThe scree plot tells you exactly how many components to keep
Read component loadings to name what each PC actually means
6 minRead component loadings to name what each PC actually means
Project onto PC1 and PC2 to see clusters your eyes can't see in 30 dimensions
7 minProject onto PC1 and PC2 to see clusters your eyes can't see in 30 dimensions
Reconstruction error tells you what your reduced data forgot
7 minReconstruction error tells you what your reduced data forgot
Phase 3Where PCA Earns Its Keep — and Where It Quietly Fails
Spot when PCA fails — and reach for t-SNE
Your boss says 'just run PCA on the user clickstream data'
7 minYour boss says 'just run PCA on the user clickstream data'
The recommendation team wants PCA on 100K-feature sparse vectors
7 minThe recommendation team wants PCA on 100K-feature sparse vectors
Stakeholder asks: 'What does PC1 actually mean?'
7 minStakeholder asks: 'What does PC1 actually mean?'
Three datasets where PCA is the wrong reach — and one cheat sheet to remember it
8 minThree datasets where PCA is the wrong reach — and one cheat sheet to remember it
Phase 4Compress 50 Features to 5 and Defend Exactly What You Lost
Compress 50 features to 5 and defend the loss
Compress 50 features to 5 — and write the one-page defense
25 minCompress 50 features to 5 — and write the one-page defense
Frequently asked questions
- What does PCA actually do in plain language?
- This is covered in the “PCA: Dimensionality Reduction from Eigenvectors” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- Why are principal components the eigenvectors of the covariance matrix?
- This is covered in the “PCA: Dimensionality Reduction from Eigenvectors” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- How do I pick the right number of principal components?
- This is covered in the “PCA: Dimensionality Reduction from Eigenvectors” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- When should I use PCA versus t-SNE or UMAP?
- This is covered in the “PCA: Dimensionality Reduction from Eigenvectors” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
- Why do I have to scale features before running PCA?
- This is covered in the “PCA: Dimensionality Reduction from Eigenvectors” learning path. Start with daily 5-minute micro-lessons that build from fundamentals to hands-on application.
Related paths
🔵Learn Set Theory Basics: The Language Every Math Class Assumes
Stop squinting at ∪, ∩, and ∁ — shade Venn diagrams first, then translate them into clean notation, until you can model anything from music genres to probability events as sets you can actually picture.
🐍Python Decorators Introduction
Build one mental model for Python decorators that covers closures, argument passing, functools.wraps, and stacking — then ship a working caching or logging decorator from scratch in under 30 lines.
🦀Rust Lifetimes Explained
Stop reading `'a` as line noise and start reading it as scope arithmetic — one failing snippet at a time — until you can thread lifetimes through a small parser or iterator adapter without fighting the borrow checker.
📈Big O Intuition
Stop treating Big O as math you memorized for an interview — build the intuition to spot O(n²) disasters, pick the right data structure without thinking, and rewrite a slow function from O(n²) to O(n) in under five minutes.