Structure-preserving Perturbations Research Articles

First order structure-preserving perturbation theory for eigenvalues of symplectic matrices: Part II

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied.

Perturbation theory of structured matrix pencils with no spillover

This paper aims to determine unstructured or structured perturbations of a matrix pencil such that the perturbed matrix pencil reconstructs a given deflating pair without changing the complementary deflating pair of the unperturbed matrix pencil. If the latter is unknown, then it is referred to as no spillover updating. If the complementary deflating pair of the given pencil is known, then all possible unstructured perturbations are obtained. However, if the complementary deflating pair of the unperturbed pencil is unknown, then we obtain a class of structure-preserving perturbations of the structured pencil. Finally, these results are utilized to obtain parametric solutions of the model updating problem with no spillover for various structured matrix pencils.

The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to the manifold of similarly structured singular matrices, and with determining the closest matrix to $T$ in this manifold. Explicit formulas are presented, exploiting the analysis of the sensitivity of the spectrum of $T$ with respect to structure-preserving perturbations of its entries.

Preserving spectral properties of structured matrices under structured perturbations

This paper is devoted to the study of preservation of eigenvalues, Jordan structure and complementary invariant subspaces of structured matrices under structured perturbations. Perturbations and structure-preserving perturbations are determined such that a perturbed matrix reproduces a given subspace as an invariant subspace and preserves a pair of complementary invariant subspaces of the unperturbed matrix. These results are further utilized to obtain structure-preserving perturbations which modify certain eigenvalues of a given structured matrix and reproduce a set of desired eigenvalues while keeping the Jordan chains unchanged. Moreover, a no spillover structured perturbation of a structured matrix is obtained whose rank is equal to the number of eigenvalues (including multiplicities) which are modified, while preserving the rest of the eigenvalues and the corresponding Jordan chains which need not be known. The specific structured matrices considered in this paper form the Lie algebra and Jordan algebra corresponding to an orthosymmetric scalar product.

Low-Rank Perturbation of Regular Matrix Pencils with Symmetry Structures

The generic change of the Weierstraß canonical form of regular complex structured matrix pencils under generic structure-preserving additive low-rank perturbations is studied. Several different symmetry structures are considered, and it is shown that for most of the structures, the generic change in the eigenvalues is analogous to the case of generic perturbations that ignore the structure. However, for some odd/even and palindromic structures, there is a different behavior for the eigenvalues 0 and \(\infty \), respectively, \(+1\) and \(-1\). The differences arise in those cases where the parity of the partial multiplicities in the perturbed matrix pencil provided by the generic behavior in the general structure-ignoring case is not in accordance with the restrictions imposed by the structure. The new results extend results for the rank-1 and rank-2 cases that were obtained in Batzke (Linear Algebra Appl 458:638–670, 2014, Oper Matrices 10:83–112, 2016) for the case of special structure-preserving perturbations. As the main tool, we use decompositions of matrix pencils with symmetry structure into sums of rank-1 matrix pencils, as those allow a parametrization of the set of matrix pencils with a given symmetry structure and a given rank.

On propagation of linear waves for model of equatorial wave–current interactions

Of current mathematical and physical interest are equatorial waves, in the setting of an azimuthal two-dimensional inviscid flow in a two-layer fluid with a flat bed and a free surface. The Hamiltonian formulation for nonlinear governing equations is adequate for structure-preserving perturbations, at the linear and at the nonlinear level have been derived. Since the ocean energy is concentrated in the long-wave propagation modes, it is of interest to pursue an in-depth analysis of long waves. In the present paper we will consider the process in the absence of vorticity.

First Order Structure-Preserving Perturbation Theory for Eigenvalues of Symplectic Matrices

A first order perturbation theory for eigenvalues of real or complex $J$-symplectic matrices under structure-preserving perturbations is developed. The main tools in the analysis are structured can...

Equatorial Wave\u2013Current Interactions

We study the nonlinear equations of motion for equatorial wave–current interactions in the physically realistic setting of azimuthal two-dimensional inviscid flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface. We derive a Hamiltonian formulation for the nonlinear governing equations that is adequate for structure-preserving perturbations, at the linear and at the nonlinear level. Linear theory reveals some important features of the dynamics, highlighting differences between the short- and long-wave regimes. The fact that ocean energy is concentrated in the long-wave propagation modes motivates the pursuit of in-depth nonlinear analysis in the long-wave regime. In particular, specific weakly nonlinear long-wave regimes capture the wave-breaking phenomenon while others are structure-enhancing since therein the dynamics is described by an integrable Hamiltonian system whose solitary-wave solutions are solitons.

Stability Radii for Linear Hamiltonian Systems with Dissipation Under Structure-Preserving Perturbations

Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical systems. One of the major advantages of the DH formulation is that system properties are encoded in an algebraic way. For instance, the algebraic structure of DH systems guarantees that the system is automatically stable. In this paper the question is discussed when a linear constant coefficient DH system is on the boundary of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues, or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well understood. In this paper, explicit formulas for this distance under structure-preserving perturbations are determined. It is also shown (via numerical examples) that under structure-preserving perturbations the asymptotical stability of a DH system is much more robust than under general perturbations, since the distance to instability can be much larger when structure-preserving perturbations are considered.

Structured maximal perturbations for Hamiltonian eigenvalue problems

We discuss the effect of structure-preserving perturbations on complex or real Hamiltonian eigenproblems and characterize the structured worst-case effect perturbations. We derive explicit expressions for the maximal Hamiltonian perturbations.

Lipschitz stability of canonical Jordan bases of H-selfadjoint matrices under structure-preserving perturbations

In this paper we study Jordan-structure-preserving perturbations of matrices selfadjoint in the indefinite inner product. The main result of the paper is Lipschitz stability of the corresponding affiliation matrices. The result can be reformulated as Lipschitz stability, under small perturbations, of canonical Jordan bases (i.e., eigenvectors and generalized eigenvectors enjoying a certain flipped orthonormality relation) of matrices selfadjoint in the indefinite inner product. The proof relies upon the analysis of small perturbations of invariant subspaces, where the size of a permutation of an invariant subspace is measured using the concepts of a gap and of a semigap.

Structured condition numbers and backward errors in scalar product spaces

The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid tolinear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of ascalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.Structured backward errors for the approximate solution of linear systems are also considered. Conditions are given for the structured backward error to be finite. For Lie and Jordan algebras it is proved that, whenever the structured backward error is finite, it is within a small factor of or equal to the unstructured one. The same conclusion holds for orthogonal and unitary structures but cannot easily be extended to other matrix groups. This work extends and unifies earlier analyses.