Mathematics · advanced
Eigenvalues & Matrix Decomposition
Eigenvectors as directions preserved by a transformation; eigenvalues as stretch factors; SVD as the universal factorization.
Mental model
Most matrices scramble space, but eigenvectors are the special directions that only get stretched, not rotated. SVD finds the best low-rank approximation of any matrix — the backbone of PCA and many compression tricks.
How to study Eigenvalues & Matrix Decomposition
Begin by restating the mental model in your own words, then connect it to a concrete system you have built or operated. Name the mechanism, the constraint it addresses, and the trade-off it introduces. Use Computational Linear Algebra (fast.ai) to check details, but close the source before writing your explanation. Retrieval is the learning step; rereading is only preparation.
Next, compare Eigenvalues & Matrix Decomposition with Linear Regression, Embeddings (Transformer). Ask what changes in correctness, latency, resource use, operability, and failure recovery. Complete Interpret an eigenvalue and preserve the command, input, output, and one failed attempt as evidence. Finish by explaining the idea without jargon to someone who has not studied the track.
Proof of understanding
- Explain the mechanism from first principles and identify the state it reads or changes.
- Give one situation where the concept is the right choice and one where it is not.
- Predict a realistic failure mode before running the drill, then compare the prediction with evidence.
- Connect the result to a roadmap or build artifact instead of treating the concept as isolated trivia.
Where it matters
PCA for dimensionality reduction, PageRank, spectral clustering, low-rank embeddings.
Common mistakes
- Thinking every matrix has real eigenvalues
- Confusing eigenvectors of A with eigenvectors of A^T
- Using PCA without centering the data first
Learn from primary sources
Practice and explain it back
Interpret an eigenvalue
Matrix A = [[2, 1], [1, 2]] has eigenvector v = [1, 1] with eigenvalue λ = 3. Explain in one sentence what Av looks like geometrically. What is the other eigenvalue of this symmetric matrix?
Expected evidence: Av = 3v — v is only stretched by 3, not rotated. The other eigenvalue is 1 (eigenvector [1, -1]).
Open the interactive drill →Review prompts
- What is an eigenvector, and why does SVD matter?
Build evidence
PCA from scratch
Center data, compute covariance matrix, eigen-decompose, project to 2D. Visualize.
- Top-2 eigenvectors explain ≥70% variance on a test dataset
- 2D scatter of projected points
- One sentence: what the first principal component 'means' for your data