Eigenlimin

Eigenlimin is a mathematical concept used to describe the asymptotic lower bounds of eigenvalues in parameterized families of matrices or operators. It is not a standard widely adopted term, but it can be defined in contexts where one studies how eigenvalues evolve along a sequence or family.

Definition and notation. Suppose {A_n} is a sequence of square Hermitian (or real symmetric) matrices with eigenvalues λ_1(n) ≤ ... ≤ λ_n(n). For a fixed index k, the eigenlimin of order k along the sequence is defined as the liminf of the k-th eigenvalue: eigenlimin_k = liminf_{n→∞} λ_k(n). More generally, if the sequence of eigenvalues has limit points, eigenlimin_k can be taken as the infimum of those limit points. The concept extends to families of operators in which each operator has a discrete spectrum of isolated eigenvalues.

Relation to limits. If A_n converges in norm to a limit operator A and the eigenvalues λ_k(n) converge to a limit μ_k (for simple eigenvalues or under suitable non-crossing conditions), then eigenlimin_k equals μ_k. Unitary similarity does not affect eigenvalues, so eigenlimin is invariant under such transformations.

Examples. For A_n = diag(n, 0, ..., 0), the largest eigenvalue grows without bound, giving eigenlimin_1 = ∞, while eigenlimin_2 = 0. For A_n = diag(1, 1/n, 1/n^2, ...), eigenlimin_1 = 1 and eigenlimin_k = 0 for k ≥ 2.

Applications. The eigenlimin concept can aid in trimming asymptotic analyses in numerical linear algebra, stability assessments of discretizations, and spectral theory where long-run lower bounds of eigenvalues are of interest. See also liminf and spectral theory.