Eigenvalue

An eigenvalue of a linear transformation is a scalar by which the transformation scales a nonzero vector that it maps to a multiple of itself.

Eigenvector: A nonzero vector that a linear transformation maps to a scalar multiple of itself. Its direction stays fixed while the eigenvalue gives its scaling factor.
Matrix exponential: The matrix-valued function defined by the power series for the exponential, used to solve linear systems. Eigenvalues determine growth or decay rates in continuous-time linear dynamics.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. Eigenvalues describe how a linear transformation acts along special vector directions.
Singular value: A nonnegative square root of an eigenvalue of a matrix's conjugate-transpose product. Unlike eigenvalues, singular values measure stretching without requiring invariant directions.
Characteristic polynomial: A polynomial whose roots are the eigenvalues of a square matrix, counted with algebraic multiplicity. Its roots locate the scalars satisfying the eigenvalue equation.
Principal component analysis: A statistical method that projects data onto directions of greatest variance. Covariance-matrix eigenvalues quantify variance along principal components.
Vector space: A set equipped with vector addition and scalar multiplication satisfying the vector-space axioms. Eigenvalues and eigenvectors are defined for linear maps on vector spaces.
Spectral radius: The largest absolute value among a matrix's eigenvalues. It compresses the full eigenvalue set into one measure of dominant magnitude.
Eigenspace: The subspace consisting of zero and all eigenvectors associated with one eigenvalue. It gathers every direction sharing this eigenvalue.
PageRank: A method for ranking web pages using the stationary distribution of a link-following process. Its rankings arise from an eigenvector of a modified web-link matrix.

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