advanced practice · coding problem
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?
The mechanism
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.
Expected evidence
Av = 3v — v is only stretched by 3, not rotated. The other eigenvalue is 1 (eigenvector [1, -1]).
Try this drill in the practice workspace →
Hints when you need them
- Eigenvectors are fixed directions under the map.
- Trace = sum of eigenvalues for 2×2.
Watch for these mistakes
- Thinking every matrix has real eigenvalues
- Confusing eigenvectors of A with eigenvectors of A^T
- Using PCA without centering the data first