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Walk through every arithmetic step of PCA on a concrete 4-point dataset: centering, forming the 2×2 covariance matrix, solving the characteristic quadratic, projecting scores, and verifying Pythagorean error.
Oct 26, 2026
Visualize Lagrange multipliers without black-box calculus: see why the highest variance contour must be tangent to the unit circle, forcing their perpendicular gradients 2Cu and 2u to align.
Oct 26, 2026
Understand Av = λv from first principles: why almost every vector gets knocked off its span during a matrix transformation, while eigenvectors stay locked on their line of action.
Oct 26, 2026
A complete 6-chapter, 24-article course uniting the Top-Down optimization view of Principal Component Analysis (minimizing orthogonal reconstruction error via the Pythagorean theorem) with the Bottom-Up geometric view of linear algebra (matrices as space transformations, eigenvectors as invariant axes, covariance ellipses, Lagrange multipliers, and SVD).
Oct 26, 2026
Substitute Cu = λu into V(u) = uᵀCu to prove that the Lagrange multiplier λ is not a mere mathematical artifact—it is literally the variance of the data along that eigenvector.
Oct 26, 2026
Discover how X = UΣVᵀ connects singular values σᵢ to eigenvalues λᵢ = σᵢ²/n, and why scikit-learn and NumPy compute PCA via SVD to prevent condition-number squaring and floating-point precision loss.
Oct 26, 2026
Calculate sample mean vectors, translate multivariate datasets so their centroid sits at the origin, and diagnose how uncentered data distorts subspace projections.
Oct 26, 2026
Prove in three clean lines of algebra why symmetric matrices never shear space obliquely—guaranteeing real eigenvalues and mutually perpendicular principal components.
Oct 26, 2026
Build bottom-up geometric intuition for linear algebra: see matrices as space-warping machines, eigenvectors as stationary axes of pure stretch, and determinants as volume scaling factors.
Oct 26, 2026
Stop memorizing the characteristic polynomial as a magic recipe: see why subtracting λI shifts the transformation until space squishes flat into a lower dimension.
Oct 26, 2026
Discover the geometric heart of PCA: how the right triangle between a data point, its shadow, and its orthogonal residual proves that minimizing compression error is mathematically identical to maximizing variance.
Oct 26, 2026
Derive the bridge formula V(u) = uᵀCu showing how the covariance matrix acts as a directional variance meter for any unit vector u in space.
Oct 26, 2026
