Eigenvectors at work: Markov chains, PCA, SVD

Pick 2–6 states, name them, and type the transition matrix. Column j lists where you go from state j (the book’s convention: columns sum to 1 and distributions update as x ↦ Px; Transpose converts a row-stochastic matrix). The state diagram shows each arrow’s probability, and the day slider fills each state by the chance of being there that day.

Enter a 2 × 2 matrix and a start vector. The picture shows v₀ … v_k, each renormalised, swinging onto the dominant eigenvector; the chart compares the error with |λ₂/λ₁|ᵏ, and the table gives the Rayleigh quotient vᵀAv converging to λ₁. Below, Aᵏ is computed exactly (repeated squaring with big integers when the entries are whole numbers) and as P Dᵏ P⁻¹ from the eigenvalues and eigenvectors.

Fₙ exactly as the corner of [[1,1],[1,0]]ⁿ by repeated squaring (every step listed), against Binet’s formula evaluated as the book’s code does, round(φn / 50.5) in 64-bit floating point. Matching leading digits are green, the rest red; the chart counts the agreeing digits for every n up to 1474, after which φⁿ overflows. At n = 100 the two numbers share 14 leading digits.

A projection matrix by age class: births along the top row, survival below the diagonal. The tool runs the population year by year, shows each class and the total (log scale optional), the share of each class settling on the stable age distribution (the dominant eigenvector), the growth rate λ, and the harvest h = 1 − 1/λ that would hold the population steady. The harvest slider applies (1 − h)L.

Drag on the scatter plot (or use ← →) to turn a line through the centre. Every point’s projection is ticked onto the line and the variance of the projections, uᵀCu, updates live; the dial plots that variance for every direction. Hunt for the maximum, then press Reveal PC1: the first principal component, the top eigenvector of the covariance matrix, lands on your best line. Sliders change the correlation, the noise and the number of points, and the eigenvalue ratio tracks how elongated the cloud is. Height & arm span is the book’s 500 people (scaled so the covariance matrix is exactly the book’s); switching height to millimetres shows mixed units hijacking PC1.

Choose the book’s 200 × 200 image (smooth waves, a disc and a little noise), another built-in sample, or your own photo — it is turned into greyscale and processed only in this page. The SVD is computed by one-sided Jacobi rotations in a background Web Worker. The rank k slider redraws the rank-k image beside the original, with the numbers stored, k(m + n + 1) against mn, and the relative error, both measured and from the singular-value tail (Eckart–Young). The singular values are plotted on a log scale with your cut-off, and the first eight rank-1 layers σᵢuᵢvᵢᵀ are shown as images — stripes that add up to the picture.

The last panel takes any 2 × 2 matrix and shows the unit circle after Vᵀ, Σ Vᵀ and U Σ Vᵀ (the step slider), with v₁ and v₂ and where they go, σ₁σ₂ = |det A|, and U and V.

A term–document matrix (rows are terms, columns documents), a query, and the number of concepts k. The table compares each document’s raw cosine with the query against its cosine in the k-dimensional concept space; the plot shows documents, terms and the query in the first two concepts. With the book’s five documents, “car” finds the document that only says “automobile”.

Related: the Matrix calculator (eigenvalues, eigenvectors, decompositions), Matrix transformations (eigenvectors and the SVD as a moving picture), Linear systems, visually, Statistics, and The maths of machine learning.

Companion to chapter 28, Eigenvectors, Bases, and Applications, of Mathematics, Reconnected: Fibonacci via matrix powers, Markov chains, PageRank, PCA and the SVD.

A link opens the tool with the given settings: /tools/pca-svd/?<parameters>#<tab>, where the tab is markov, power, fib, leslie, pca, svd or lsi. Share link (top right) builds one from whatever is on screen. Matrices are written row by row: entries separated by commas, rows by semicolons (0.9,0.5;0.1,0.5); fractions such as 1/3 work. A + typed straight into a link means a space to browsers, so leave it out (3 not +3) or write %2B.

Presets for example=:

For example, https://geekychris.github.io/tools/pca-svd/?example=ch28-pagerank#markov opens the PageRank example, and https://geekychris.github.io/tools/pca-svd/?P=0.7,0.4;0.3,0.6&x0=0,1&names=Work%20out,Skip#markov the gym chain starting from a skipped day.