248 research outputs found
Perturbation bounds for isotropic invariant subspaces of skew-Hamiltonian matrices
We investigate the behavior of isotropic invariant subspaces of skew-Hamiltonian matrices under structured perturbations. It is shown that finding a nearby subspace is equivalent to solving a certain quadratic matrix equation. This connection is used to derive meaningful error bounds and condition numbers that can be used to judge the quality of invariant subspaces computed by strongly backward stable eigensolvers
Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications
Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described
Structured condition numbers for invariant subspaces
Invariant subspaces of structured matrices are sometimes better conditioned with respect to structured perturbations than with respect to general perturbations. Sometimes they are not. This paper proposes an appropriate condition number c(S), for invariant subspaces subject to structured perturbations. Several examples compare c(S) with the unstructured condition number. The examples include block cyclic, Hamiltonian, and orthogonal matrices. This approach extends naturally to structured generalized eigenvalue problems such as palindromic matrix pencils
Structured Eigenvalue Problems
Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices
Broken translation invariance in quasifree fermionic correlations out of equilibrium
Using the C* algebraic scattering approach to study quasifree fermionic
systems out of equilibrium in quantum statistical mechanics, we construct the
nonequilibrium steady state in the isotropic XY chain whose translation
invariance has been broken by a local magnetization and analyze the asymptotic
behavior of the expectation value for a class of spatial correlation
observables in this state. The effect of the breaking of translation invariance
is twofold. Mathematically, the finite rank perturbation not only regularizes
the scalar symbol of the invertible Toeplitz operator generating the leading
order exponential decay but also gives rise to an additional trace class Hankel
operator in the correlation determinant. Physically, in its decay rate, the
nonequilibrium steady state exhibits a left mover--right mover structure
affected by the scattering at the impurity.Comment: 30 pages, 4 figure
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