Michael Karow

Structured Eigenvalue Condition Numbers

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, skew-symmetric, 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.

Perturbation theory for Hamiltonian matrices and the distance to bounded-realness

Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.

On the computation of structured singular values and pseudospectra

Structured singular values and pseudospectra play an important role in assessing the properties of a linear system under structured perturbations. This paper discusses computational aspects of structured pseudospectra for structures that admit an eigenvalue minimization characterization, including the classes of real, skew-symmetric, Hermitian, and Hamiltonian perturbations. For all these structures we develop algorithms that require O (n2) operations per grid point, combining the Schur decomposition with a Lanczos method. These algorithms form the basis of a graphical Matlab interface for plotting structured pseudospectra.

Self-adjoint operators and pairs of Hermitian forms over the quaternions

Abstract We classify self-adjoint operators and pairs of Hermitian forms over the real quaternions by providing canonical matrix representations. In the preliminaries we discuss the Jordan canonical form theorem for quaternionic linear endomorphisms.

On the structured distance to uncontrollability

This article is concerned with the structured distance to uncontrollability of a linear time-invariant system and relates this concept to a variation of the μ-value. The developed framework is applied to derive computational expressions for the class of real perturbations as well as for Hermitian, symmetric, and skew-symmetric perturbations in a relatively simple manner. Examples demonstrate that the structured distance can differ from the standard, unstructured distance to uncontrollability by an arbitrary amount. It is also shown how systems of higher order can be addressed.

μ-VALUES AND SPECTRAL VALUE SETS FOR LINEAR PERTURBATION CLASSES DEFINED BY A SCALAR PRODUCT *

We study the variation of the spectrum of matrices under perturbations which are self- or skew-adjoint with respect to a scalar product. Computable formulas are given for the associated μ-values. The results can be used to calculate spectral value sets for the perturbation classes under consideration. We discuss the special case of complex Hamiltonian perturbations of a Hamiltonian matrix in detail.

Structured Eigenvalue Backward Errors of Matrix Pencils and Polynomials with Hermitian and Related Structures

We derive a formula for the backward error of a complex number

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

\lambda

Structured Pseudospectra for Small Perturbations

when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian,

EIGENVALUE CONDITION NUMBERS AND A FORMULA OF BURKE, LEWIS AND OVERTON ∗

*