Eigenvalues and eigenvectors

For a linear transformation, an eigenvector is a nonzero vector whose direction is unchanged, and its eigenvalue is the factor by which it is scaled.

Characteristic polynomial: The polynomial det(λI − A) whose roots are the eigenvalues of a square matrix A. Its roots identify the scaling factors that allow nonzero vectors to retain their directions.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. Eigenvectors describe special directions preserved by this kind of map.
Principal component analysis: A statistical method that projects data onto directions of greatest variance. Eigenvectors of the covariance matrix give the principal directions; eigenvalues measure their variances.
Singular value decomposition: A factorization of any real or complex matrix into two unitary factors and a diagonal matrix of nonnegative values. Unlike eigen-decomposition, it applies to rectangular matrices and separates left and right directions.
Eigenspace: The subspace consisting of zero and all eigenvectors associated with one eigenvalue. It collects every invariant direction sharing the same scaling factor.
Matrix: A rectangular array of numbers representing a linear map after bases are chosen. Matrix equations provide the standard way to calculate eigenvalues and eigenvectors.
Vibrational mode: A pattern of oscillation in which a system’s components move at one shared frequency. Normal modes arise as eigenvectors of mechanical systems, with eigenvalues related to squared frequencies.
Schur decomposition: A unitary similarity factorization of a complex square matrix into upper-triangular form. It exists for every complex square matrix, even when a full eigenvector basis does not.
Geometric multiplicity: The dimension of an eigenvalue’s eigenspace. It counts the independent eigenvector directions available for a particular eigenvalue.
Vector space: A set equipped with vector addition and scalar multiplication satisfying the vector-space axioms. Eigenvectors are nonzero elements of a vector space acted on by a linear map.

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