Eigenvector

An eigenvector of a linear transformation is a nonzero vector that the transformation scales without changing its direction. The scale factor is its eigenvalue.

Eigenvalue: A scalar by which a linear transformation scales an eigenvector. It gives the scale factor paired with each preserved direction.
Linear transformation: A map between vector spaces that preserves vector addition and scalar multiplication. Eigenvectors describe the special directions of such a map.
Principal component analysis: A statistical method that projects data onto directions of greatest variance. Its principal directions are eigenvectors of the data covariance matrix.
Singular vector: A vector mapped by a matrix to a scalar multiple of a corresponding vector under the transpose action. Unlike eigenvectors, singular vectors are defined for rectangular matrices too.
Rotation matrix: A matrix representing a rotation of a coordinate plane or higher-dimensional Euclidean space. A nontrivial planar rotation has no real eigenvector, despite preserving lengths.
Eigenspace: The subspace consisting of all eigenvectors for an eigenvalue, together with the zero vector. It gathers every preserved direction associated with the same scaling factor.
Matrix: A rectangular array of numbers that can represent a linear map between finite-dimensional coordinate spaces. Matrix multiplication provides the standard computational form of the eigenvector equation.
Markov chain: A stochastic process whose next state depends on its current state through transition probabilities. A stationary distribution is an eigenvector of the transition matrix with eigenvalue one.
Generalized eigenvector: A vector that satisfies a higher-power eigenvalue equation when ordinary eigenvectors do not span the space. It extends eigenvector methods to matrices that cannot be diagonalized.
Reflection matrix: A linear transformation that reflects vectors across a line or plane through the origin. Vectors along and perpendicular to its mirror plane are eigenvectors.

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