Eigenvalue problem

An eigenvalue problem seeks parameter values for which a linear operator equation has nonzero solutions. Those values are eigenvalues, and the corresponding solutions are eigenvectors or eigenfunctions.

Eigenvalue: A scalar associated with a linear operator when a nonzero vector or function is mapped to its scalar multiple. It is the parameter value the problem seeks.
Linear operator: A map between vector spaces that preserves vector addition and scalar multiplication. The operator is the central object whose special parameter values are sought.
Sturm–Liouville theory: A theory of second-order linear differential operators with boundary conditions and weighted orthogonal eigenfunctions. It provides a canonical framework for many differential eigenvalue problems.
Generalized eigenvalue problem: A problem seeking nonzero vectors satisfying Av = λBv for given linear maps A and B. It replaces the identity operator with a second operator or matrix.
Eigenvector: A nonzero vector that a linear transformation maps to a scalar multiple of itself. Finite-dimensional problems seek vectors satisfying the operator equation.
Vector space: A set whose elements can be added and multiplied by scalars while satisfying the vector-space axioms. Eigenvectors and operator actions are defined within vector spaces.
Vibrating string: A stretched string whose transverse motion is governed by a wave equation. Its allowed standing-wave frequencies arise from a boundary-value eigenproblem.
Singular value decomposition: A matrix factorization into orthogonal or unitary transformations and a diagonal matrix of nonnegative singular values. It analyzes arbitrary rectangular matrices where ordinary eigenvalue analysis may not apply.
Eigenfunction: A nonzero function mapped to a scalar multiple of itself by a linear operator. Differential-operator problems seek functions satisfying this relation.
Identity operator: The linear operator that maps every vector to itself. Subtracting a scalar multiple of it forms the operator equation for eigenvalues.

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