On Matrix Nearness Problems: Distance to Delocalization

@article{Kostic2015OnMN,
  title={On Matrix Nearness Problems: Distance to Delocalization},
  author={Vladimir Kostic and Agnieszka Miedlar and Jeroen J. Stolwijk},
  journal={SIAM J. Matrix Anal. Appl.},
  year={2015},
  volume={36},
  pages={435-460}
}
In this paper we introduce a new matrix nearness problem that is intended to generalize the distance to instability. Due to its applicability in analyzing the robustness of eigenvalues with respect to the arbitrary localization sets (domains) in the complex plane, we call it the distance to delocalization. For the open left half-plane or the unit disk, the distance to the nearest unstable matrix is obtained as a special case. Following the theoretical framework of Hermitian functions and the… 

Nearest $\varOmega $-stable matrix via Riemannian optimization

TLDR
The task of finding the nearest $\Omega$-stable matrix to a certain matrix $A$ is described as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices, and the resulting algorithm is remarkably fast on small-scale and medium-scale matrices.

Nearest Ω-stable matrix via Riemannian optimization

TLDR
The task of finding the nearest Ω-stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Ω is described as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices.

New pseudospectra localizations with application in ecology and vibration analysis

TLDR
This paper derives some new pseudospectra localization sets, and compares them with the known ones, using relevant numerical examples arising in applications in ecology and vibration analysis.

References

SHOWING 1-10 OF 43 REFERENCES

Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking

TLDR
Using the property that the pseudospectrum is determined via perturbations by rank-1 matrices, differential equations on the manifold of normalized rank- 1 matrices whose solutions tend to the critical rank-2 perturbation associated with the extremal points of (locally) maximum real part and modulus are derived.

A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices

We describe a bisection method to determine the 2-norm and Frobenius norm distances from a given matrix A to the nearest matrix with an eigenvalue on the imaginary axis. If A is stable in the sense

On general principles of eigenvalue localizations via diagonal dominance

TLDR
A concept of DD-type classes of matrices is introduced and how to construct eigenvalue localization sets are shown and obtained principles can be used to construct and use novel Geršgorin-like localization areas.

The calculation of the distance to a nearby defective matrix

TLDR
A new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented by an extension of the implicit determinant method introduced by Spence and Poulton in (2005).

Low rank differential equations for Hamiltonian matrix nearness problems

TLDR
A characterization of optimal perturbations is obtained that turn out to be of low rank and are attractive stationary points of low-rank differential equations that are derived from the Hamiltonian matrix nearness problems.

Subspace Methods for Computing the Pseudospectral Abscissa and the Stability Radius

TLDR
This paper proposes to combine a linearly converging iterative method for computing the pseudospectral abscissa and its variants with subspace acceleration, and observes local quadratic convergence and proves local superlinear convergence of the resulting subspace methods.

Fast Algorithms for the Approximation of the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix

TLDR
New algorithms based on computing only the spectral abscissa or radius of a sequence of matrices are presented, generating a sequences of lower bounds for the pseudospectral absc Melissa or radius, proving a locally linear rate of convergence for $\varepsilon$ sufficiently small.

Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

Two useful measures of the robust stability of the discrete-time dynamical system xk+1 = Axk are the � -pseudospectral radius and the numerical radius of A. The � -pseudospectral radius of A is the

MATRIX NEARNESS PROBLEMS AND APPLICATIONS

TLDR
A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability.