This is the shared mathematical companion for Eigen Times and Eigen Hacks. It accompanies Eigen Times: A Newspaper in the Eigenbasis of the News, which states the methods compactly and reports their results. Here we slow down. A reader needs arithmetic, fractions, and a willingness to work through small lists of numbers. We explain the other tools when they become necessary: vectors, matrices, logarithms, covariance, eigenvectors, derivatives, and the singular value decomposition. The aim is to understand what each calculation measures, why we want it, and how to reproduce it.
The same mathematics can organize newspaper articles and technology discussions. Their data distributions need not be the same. A similarity threshold measured on one archive does not automatically work on another. Historical corpus counts and threshold observations in this book retain their original dates and source context; they are not current site counters or new measurements of Eigen Hacks. Synthetic examples, by contrast, are small teaching inputs that the reader can change.
A chapter first explains the problem in words, introduces the necessary terms, and then works through a calculation. Read the intermediate arithmetic before the compact formula. When a formula still looks dense, identify what goes in, what comes out, and the size of each object. The upfront guide and final glossary provide a second route through the terminology; the subject index links back to explanations.
Every matrix in the original figures is drawn as a grid of coloured squares: blue for positive entries, rust for negative, darker for larger. The figure inputs and dated aggregate observations remain frozen. The complete Python and native OCaml notebooks contain this text, all nineteen figures, and executable lessons interleaved with the chapters. Their cells calculate the worked examples from declared inputs, test identities and reference answers, and finish with matching numerical receipts. Neither notebook needs access to the production database or a private archive. The public teaching repository is querygraph/eigenmath; a delivered study bundle also contains the exact notebook editions and reading copies.
Version 1.1.2. This patch expands compressed teaching steps throughout the shared companion. It introduces the meaning of a residual before measuring its length, develops differentiation from a small change in a scalar function before using it for minimization, and unrolls normalization, covariance, decomposition, rotation, statistical thresholds, and streaming sums. It adds an upfront notation guide, a glossary, and a linked index, and repairs clipped or overlapping figure annotations. One clustering sentence is corrected to agree with its existing frozen fixture: a threshold of 0.9 leaves one linked pair on 24 February 2022. No archive is refitted and no historical threshold or measurement is replaced.