Eigen Times starts from a claim about news: that most of it recurs. Wars, scandals, market shocks, elections, disasters happen again and again with different names attached. If that is true, then the enormous variety of news articles should be mostly explained by a small number of recurring directions, and each day’s news should be describable as “so much of this recurring thing, so much of that one, plus a remainder those directions do not capture.”
Linear algebra gives that sentence a precise meaning. We will build the meaning before using the machinery. A scalar is one number. A vector is an ordered list of numbers: in a two-coordinate illustration, says three units in the first coordinate and one in the second. Order matters; exchanging the entries changes the vector. A nonzero vector points from the origin, where every coordinate is zero, in a particular direction.
For a first example, suppose our representation records only the horizontal direction . Taking three times that direction produces . This is a reconstruction: the part of the observation that our chosen direction can rebuild. Subtract reconstruction from observation, coordinate by coordinate:
That remainder is the residual. It records what the reconstruction missed. A residual is a vector before it is a length or a score; here its first entry is zero and its second entry is one. The later measurement chapter will explain exactly how to measure its size. For now, the decomposition is visible: observation equals reconstructed part plus residual.
The number three is a coordinate relative to the chosen direction: it tells us how much of that direction to use. A weighted combination adds several directions after multiplying each by a scalar. All combinations obtainable from a collection of directions form their span, a subspace. Our single horizontal direction spans a line; adding a genuinely different direction can span a plane. A low-dimensional representation keeps a limited collection of such directions. The question for Eigen Times is which directions let us reconstruct recurring patterns in news while making the remainder useful to inspect.
The notebook begins with three synthetic observations, , , and . Their horizontal coordinates are 1, 2, and 3. The first two are rebuilt completely; only the third has a nonzero remainder. These are arithmetic examples, not measured levels of war or banking. Actual text representations acquire their coordinates in the next chapter.
The chapters follow the pipeline. Text becomes vectors (§2). Vectors are stacked into matrices, and matrices are multiplied (§3). The covariance of a million article vectors is computed and its eigenvectors found (§4). The singular value decomposition works directly on the data matrix and gives us latent semantic analysis (§5). The eigenvectors are oriented and rotated into axes with names (§6). A new story is projected, and what is left over is measured (§7). Then come the thresholds (§8–§9), matching axes across bases and nights (§10), streaming the calculation (§11), and the shapes of everything (§12).