Structured Eigenvalue Condition Numbers

Abstract

Abstract. This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.

Structured Eigenvalue Problems

“…Since κ struct (λ) ≤ κ(λ) holds for any struct, this implies that there is no significant difference between κ R n×n (λ) and κ(λ). Surprisingly, the same statement holds for a variety of other structures [18,55,74,107], see also Section 3. Note that (7) implies |λ − λ| κ struct (λ) E 2 for all E ∈ struct.…”

“…Analogously, any eigenvector belonging to λ(A 11 + A 12 F n ) is center-symmetric. While the structured and unstructured eigenvalue condition numbers for symmetric persymmetric matrices are the same [74,107], there can be a significant difference in the invariant subspace condition numbers. With respect to structured perturbations, the separation between λ(A 11 − A 12 F n ) and λ(A 11 + A 12 F n ), which can be arbitrarily small, does not play any role for invariant subspaces belonging to eigenvalues from one of the two eigenvalue sets [31].…”

“…Concerning structured perturbation results for symplectic matrices, see [74,116] and the references therein. If λ is a simple eigenvalue for M , then so is 1/λ.…”

“…Concerning structured perturbation results for Hamiltonian matrices, see [31,74,77,116] and the references therein. It turns out that there is no or little difference between κ(λ) and κ Hamil (λ), the (structured) eigenvalue condition numbers.…”

Structured condition numbers and backward errors in scalar product spaces

“…Equality holds since the union of the curves A + g E contains an open neighborhood in S of A (see [10] for details). An automorphism group G forms a smooth manifold.…”