Quasieigen

Quasieigen is a term used in mathematics and physics to denote objects that closely resemble eigenvalues and eigenvectors but are not exact solutions to an eigenvalue problem. It typically arises when a system is perturbed, discretized, or otherwise difficult to solve exactly, and one seeks robust approximations to the spectral data of a linear operator.

Formally, for a linear operator A on a finite-dimensional inner product space, a pair (λ, x) with x ≠ 0 is called a quasieigenpair (or quasi-eigenpair) for A if the residual r = Ax − λx is small relative to x, commonly ||Ax − λx|| ≤ ε ||x|| for a prescribed tolerance ε > 0. The scalar λ is a quasieigenvalue and the vector x a quasieigenvector. As ε → 0, quasieigenpairs converge to true eigenpairs under suitable conditions such as diagonalizability and good conditioning.

Quasieigenpairs are central in numerical linear algebra, where they arise in methods such as the Ritz method: Ritz values approximate eigenvalues and Ritz vectors approximate eigenvectors, yielding quasieigenpairs of A. For non-Hermitian or highly non-normal operators, eigenpairs can be highly sensitive to perturbations; quasieigenpairs, and the related ε-pseudospectrum, provide a more stable view of an operator’s spectral region under perturbation.

The ε-pseudospectrum of an operator is the set of λ for which (A − λI) is nearly singular, or equivalently where (A − λI) has a large inverse norm. This concept captures where quasieigenvalues may lie when the system is subject to small changes.

In physics, the notion appears as quasi-eigenstates or quasi-energy levels, used to describe resonance-like or metastable states and the spectrum of periodically driven or open systems.

See also: eigenvalue, Ritz values, perturbation theory, pseudospectrum.