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1
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The estimates of diagonally dominant degree and eigenvalues inclusion regions for the Schur complement of block diagonally dominant matrices

100%
Wang F., Sun D.
Open Mathematics
|
2015
|
tom 13
|
nr 1
EN
The theory of Schur complement plays an important role in many fields, such as matrix theory and control theory. In this paper, applying the properties of Schur complement, some new estimates of diagonally dominant degree on the Schur complement of I(II)-block strictly diagonally dominant matrices and I(II)-block strictly doubly diagonally dominant matrices are obtained, which improve some relative results in Liu [Linear Algebra Appl. 435(2011) 3085-3100]. As an application, we present several new eigenvalue inclusion regions for the Schur complement of matrices. Finally, we give a numerical example to illustrate the advantages of our derived results.
2
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Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in l 2 by the use of finite submatrices

88%
Malejki M.
Open Mathematics
|
2010
|
tom 8
|
nr 1
114-128
EN
We consider the problem of approximation of eigenvalues of a self-adjoint operator J defined by a Jacobi matrix in the Hilbert space l 2(ℕ) by eigenvalues of principal finite submatrices of an infinite Jacobi matrix that defines this operator. We assume the operator J is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the eigenvalues, numbered from 1 to N; of J by the eigenvalues of the finite submatrix J n of order n × n; where N = max{k ∈ ℕ: k ≤ rn} and r ∈ (0; 1) is arbitrary chosen. We apply this result to obtain an asymptotics for the eigenvalues of J. The method applied in this research is based on Volkmer’s results included in [23].
3
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Complexity issues for the symmetric interval eigenvalue problem

88%
Hladík M.
Open Mathematics
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2015
|
tom 13
|
nr 1
EN
We study the problem of computing the maximal and minimal possible eigenvalues of a symmetric matrix when the matrix entries vary within compact intervals. In particular, we focus on computational complexity of determining these extremal eigenvalues with some approximation error. Besides the classical absolute and relative approximation errors, which turn out not to be suitable for this problem, we adapt a less known one related to the relative error, and also propose a novel approximation error. We show in which error factors the problem is polynomially solvable and in which factors it becomes NP-hard.
4
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Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

63%
Kanigowski A., Kryszewski W.
Open Mathematics
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2012
|
tom 10
|
nr 6
2240-2263
EN
We study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.
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