What does eigenvector mean?
An eigenvector is a fundamental concept in linear algebra: a nonzero vector whose direction is unchanged — only scaled — when a linear transformation is applied to it. Formally, for a square matrix A, an eigenvector v satisfies the equation Av = λv, where λ is a scalar known as the eigenvalue. Eigenvectors reveal the essential behaviour of transformations: they identify directions that a transformation merely stretches or compresses rather than rotates. This property makes them indispensable across science and technology. In physics, they describe stationary states of quantum systems; in statistics and machine learning, they underpin techniques such as principal component analysis; in computer science, algorithms like Google's original PageRank rely on computing dominant eigenvectors. The term comes from German, where 'eigen' means 'own' or 'characteristic', reflecting the idea that these vectors belong uniquely to their transformation. Related terms such as eigenvalue, eigenfunction, and eigenspace form a family of vocabulary central to modern applied mathematics.
nounA nonzero vector that, when a linear transformation is applied to it, changes only by a scalar factor rather than in direction. The corresponding scalar multiplier is called the eigenvalue.
- A nonzero vector v such that Av = λv for some scalar λ, where A is a square linear transformation (matrix); the direction of v is preserved by A.
- By extension in related fields (quantum mechanics, data science), a characteristic mode or principal component associated with an operator or dataset.
"For the matrix, the vector (1, 2) is an eigenvector with eigenvalue 5."
"The PageRank algorithm ranks web pages using the dominant eigenvector of the link graph."
"In quantum mechanics, measurable quantities are represented by operators whose eigenvectors correspond to possible states."
Standard regular English plural; used when referring to multiple such vectors of one or more matrices or operators.
"Computing the eigenvectors of this covariance matrix reveals the principal components of the data."
The word 'eigen' is German for 'own' or 'characteristic' — so an eigenvector is literally a matrix's very own vector, one whose direction it cannot change.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of eigenvector
The term 'eigenvector' derives from the German adjective 'eigen', meaning 'own', 'characteristic', or 'proper', combined with the English word 'vector'. It entered English mathematical usage in the mid-twentieth century, translated from German terminology developed in the late nineteenth century by mathematicians such as David Hilbert, who used 'Eigenwert' (eigenvalue) in his work on integral equations. Earlier equivalent terms in English included 'characteristic vector' and 'proper vector'. Cognate technical terms sharing the same root include eigenvalue, eigenfunction, and eigenspace.
How eigenvector is actually used
Strictly a technical term from linear algebra, used almost exclusively in mathematics, physics, engineering, and data science contexts; not used figuratively in everyday English. Plural is 'eigenvectors'.
Easily confused with eigenvector
An eigenvalue is the scalar factor by which an eigenvector is stretched or shrunk; the eigenvector is the direction that remains unchanged under the transformation.
A vector is any quantity with magnitude and direction, while an eigenvector is a special vector that keeps its direction unchanged under a given linear transformation.