What does eigenvalue mean?
An eigenvalue is one of the fundamental quantities associated with a linear transformation or square matrix. Formally, it is a number λ for which the transformation acts on some nonzero vector v purely by stretching: Av = λv. The special vectors that behave this way are called eigenvectors, and together they reveal what a transformation does at its core — which directions it merely scales rather than rotates or skews. Eigenvalues appear wherever systems evolve linearly: they determine whether a structure vibrates dangerously, whether a population model grows or collapses, and how quickly iterative algorithms converge. Computing them means solving the characteristic polynomial, and for large matrices this becomes a major task in numerical linear algebra. Beyond pure mathematics, eigenvalues underpin quantum mechanics, where measurable energies correspond to them, and modern data science, where techniques like principal component analysis extract meaning from data through them. Though technical, the concept is indispensable across science and engineering.
nounA scalar value λ for which the linear equation Av = λv has a nonzero solution vector v, where A is a square matrix (or linear transformation); equivalently, a factor by which an eigenvector is stretched or shrunk under that transformation.
- A scalar λ such that a linear transformation T satisfies T(v) = λv for some nonzero vector v; in matrix terms, a root of the characteristic polynomial det(A − λI) = 0.
"The largest eigenvalue of the matrix determines how quickly the system converges."
"To analyze the stability of the differential equation, we computed the eigenvalues of the coefficient matrix."
"Google's original PageRank algorithm relies on finding the principal eigenvalue of a vast link matrix."
The standard plural follows regular English noun inflection. Because a single eigenvalue carries little information on its own, the plural form is especially common in practice.
"All the eigenvalues of this symmetric matrix are real."
Every square matrix keeps a few special directions where it behaves like simple multiplication — and the numbers it multiplies by are its eigenvalues.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of eigenvalue
The term is a partial calque from German 'Eigenwert', coined by the mathematician David Hilbert around 1904 in his work on integral equations; 'eigen' means 'own' or 'characteristic' and 'Wert' means 'value'. English speakers kept the German prefix untranslated, giving rise to the whole family of 'eigen-' terms — eigenvector, eigenfunction, eigenstate — that spread through mathematics and physics in the early twentieth century. Earlier English texts had used translations such as 'characteristic value' or 'proper value', which still survive in some older literature.
How eigenvalue is actually used
A technical term used almost exclusively in mathematics, physics, engineering, and data science. In speech it is usually pluralized or discussed collectively ('the eigenvalues'), since a single eigenvalue is rarely meaningful without its eigenvector. The German-derived prefix 'eigen-' is pronounced /ˈaɪɡən/ in English.
Easily confused with eigenvalue
An eigenvector is the direction (nonzero vector) that is only scaled by a matrix, while the eigenvalue is the scalar factor by which it is scaled.