Eigenvalue
Quantitative Finance glossary · Probability, Statistics & Linear Algebra
Definition
Scalar λ for which a square matrix boldsymbolA satisfies boldsymbolAboldsymbolv = λboldsymbolv for some nonzero eigenvector boldsymbolv; solutions of the characteristic polynomial det(boldsymbolA - λ boldsymbolI) = 0. Symmetric matrices (covariance, correlation) have real eigenvalues and an orthonormal eigenbasis. The PCA factors of the yield curve are eigenvectors of the covariance of rate changes; eigenvalues quantify variance explained per factor.
Eigenvalue in plain English
Think of a matrix as a machine that takes an arrow (a vector) and stretches, shrinks, and rotates it. For most arrows the machine spins them off in some new direction — but a few special arrows come out pointing the exact same way, only longer or shorter; those are eigenvectors, and the stretch factor λ (how many times longer) is the eigenvalue. They matter because they reveal the 'natural directions' hidden inside data: for example, when analysts study how interest rates move, the biggest eigenvalue/eigenvector pair tells them the single most common way the whole rate curve shifts, capturing most of the action in one clean pattern. In short, eigenvalues are a way to find the few directions that really drive a complicated system.
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