eigenvalueeigenvector
An eigenvalue and its corresponding eigenvector of a square matrix A are a scalar λ and a nonzero vector v that satisfy the equation Av = λv. In this relation, the vector v is stretched or compressed by A by a factor of λ and directions are preserved. The equation can be rewritten as (A − λI)v = 0, so nontrivial solutions exist only when det(A − λI) = 0. The polynomial det(A − λI) is called the characteristic polynomial, and its roots are the eigenvalues of A. The eigenvectors associated with each eigenvalue are the nonzero solutions to (A − λI)v = 0.
Eigenvalues can be repeated; their multiplicities are categorized as algebraic multiplicity (multiplicity as a root of
Computation typically proceeds by first finding the eigenvalues from det(A − λI) = 0, then solving (A − λI)v
Applications span many areas. In linear dynamical systems, eigenvalues determine stability and long-term behavior. In data