11 research outputs found
On determining the domain of the adjoint operator
A theorem that is of aid in computing the domain of the adjoint operator is
provided. It may serve e.g. as a criterion for selfadjointness of a symmetric
operator, for normality of a formally normal operator or for
--selfadjointness of an --symmetric operator. Differential operators and
operators given by an infinite matrix are considered as examples
On a class of H-selfadjont random matrices with one eigenvalue of nonpositive type
Large H-selfadjoint random matrices are considered. The matrix is assumed
to have one negative eigenvalue, hence the matrix in question has precisely one
eigenvalue of nonpositive type. It is showed that this eigenvalue converges in
probability to a deterministic limit. The weak limit of distribution of the
real eigenvalues is investigated as well
Matrix methods for Pad\'e approximation: numerical calculation of poles, zeros and residues
A representation of the Pad\'e approximation of the -transform of a signal
as a resolvent of a tridiagonal matrix is given. Several formulas for the
poles, zeros and residues of the Pad\'e approximation in terms of the matrix
are proposed. Their numerical stability is tested and compared. Methods for
computing forward and backward errors are presented
On the spectral properties of a class of -selfadjoint random matrices and the underlying combinatorics
An expansion of the Weyl function of a -selfadjoint random matrix with one
negative square is provided. It is shown that the coefficients converge to a
certain generalization of Catlan numbers. Properties of this generalization are
studied, in particular, a combinatorial interpretation is given
Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures
Rank two parametric perturbations of operators and matrices are studied in
various settings. In the finite dimensional case the formula for a
characteristic polynomial is derived and the large parameter asymptotics of the
spectrum is computed. The large parameter asymptotics of a rank one
perturbation of singular values and condition number are discussed as well. In
the operator case the formula for a rank two transformation of the spectral
measure is derived and it appears to be the t-transformation of a probability
measure, studied previously in the free probability context. New transformation
of measures is studied and several examples are presented
Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matrices
General properties of eigenvalues of as functions of
\tau\in\Comp or \tau\in\Real or \tau=\e^{\ii\theta} on the unit circle
are considered. In particular, the problem of existence of global analytic
formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues
with are discussed in detail. The following classes of matrices
are considered: complex (without additional structure), real (without
additional structure), complex -selfadjoint and real -Hamiltonian
Linear algebra properties of dissipative Hamiltonian descriptor systems
A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system
Distance problems for dissipative Hamiltonian systems and related matrix polynomials
We study the characterization of several distance problems for linear differential-algebraic systems with dissipative Hamiltonian structure. Since all models are only approximations of reality and data are always inaccurate, it is an important question whether a given model is close to a 'bad' model that could be considered as ill-posed or singular. This is usually done by computing a distance to the nearest model with such properties. We will discuss the distance to singularity and the distance to the nearest high index problem for dissipative Hamiltonian systems. While for general unstructured differential-algebraic systems the characterization of these distances are partially open problems, we will show that for dissipative Hamiltonian systems and related matrix polynomials there exist explicit characterizations that can be implemented numerically