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Prove algebraically and geometrically why the projection vector and residual vector are orthogonal, and derive the exact right-triangle identity for every observation.
Oct 26, 2026
Understand why multiplying a centered data matrix by its transpose (XᵀX) automatically computes every feature's variance on the diagonal and every pairwise covariance off the diagonal.
Oct 26, 2026
See why maximizing uᵀCu without a length constraint is an ill-posed problem that explodes to infinity, and how restricting u to the unit sphere isolates pure direction from ruler scale.
Oct 26, 2026
Calculate the total scatter (average squared Euclidean norm) of a centered dataset and prove why rotating your coordinate axes never changes total data energy.
Oct 26, 2026
Connect statistics to linear algebra: see how C = (1/n)XᵀX encodes all pairwise feature relationships, how uᵀCu measures directional variance, and why symmetric matrices always have perpendicular eigenvectors.
Oct 26, 2026
Use the matrix trace identity Trace(C) = Σλᵢ to compute explained variance ratios, read scree plots, and understand why feature standardization (z-scores) is critical when units differ.
Oct 26, 2026
Average the Pythagorean identity across an entire dataset to prove the conservation law of PCA: every unit of variance captured along the axis is a unit of reconstruction error eliminated.
Oct 26, 2026
A detailed exploration of Kernel Principal Component Analysis (Kernel PCA), explaining its necessity, the underlying 'kernel trick', and how it extends PCA to handle non-linear data structures.
Feb 25, 2026
A detailed guide on representation learning, focusing on why a line is a good representation, how to find the best proxy for a point, and a full derivation to find the optimal line that minimizes reconstruction error (PCA).
Dec 15, 2025
A detailed explanation of the Gram-Schmidt process, including its purpose, step-by-step procedure, and multiple examples from vector spaces to function spaces.
Dec 2, 2025
A comprehensive guide to NumPy array manipulation, from basic indexing and slicing to advanced techniques with fancy indexing, boolean masking, and the powerful np.where() function. Includes 20 challenging practice problems.
Nov 17, 2025
A detailed, step-by-step guide to understanding Singular Value Decomposition (SVD) from first principles, complete with geometric intuition and solved problems.
Nov 12, 2025
