Definition
An eigenvector is a vector that keeps pointing in the same direction when a transformation is applied to it. A transformation, written as a matrix (a grid of numbers that reshapes data), normally rotates and stretches vectors, but an eigenvector only gets longer or shorter, never turned off its line. The number that measures how much it stretches or shrinks is called the eigenvalue. When a system is described by a large matrix, its main eigenvector reveals the stable pattern that stays in place no matter how many times the transformation repeats.
A simple example shows why this is useful. Imagine one point joined to a hundred weak, unimportant points, and a second point joined to only five points that each carry a lot of weight. Even though the first point has far more connections, the second one scores higher, because its few connections are strong. An eigenvector captures this by weighting every connection by how important the thing on the other end is, so the quality of connections beats raw quantity.
Use Cases
Eigenvectors are used to find the most important elements inside large, connected systems. Search engines use them to rank web pages, social platforms use them to spot influential accounts, and recommendation systems use them to group similar items. In each case the eigenvector turns millions of connections into a single measure of how importance flows through the network.
Why It Matters for SEO
Eigenvectors matter for SEO because PageRank, the algorithm that helped build Google, is an eigenvector. Google treats the web as a huge matrix in which every link is a connection between pages. The eigenvector of that matrix gives each page an authority score based on how link value flows across the entire web. This is why authority depends on the structure of the whole link graph, not just on how many links a single page has.
Related Concepts
Related concepts include PageRank, Markov chains, link graphs, and centrality measures. These ideas all describe how importance is calculated and spread across a network. They share the same mathematical core: finding the stable pattern that appears when a system is applied to itself many times.