On the inertia of the block H-matrices

We give detailed discussion of a procedure for determining the robust D-stability of a 4 ×4 real matrix. The procedure begins from the Hur- witz stability criterion. The procedure is applied to two numerical examples.

SIAM Journal on Numerical Analysis, 1997

An algorithm presented in Hager [Comput. Math. Appl., 14 (1987), pp. 561-572] for diagonalizing a matrix is generalized to a block matrix setting. It is shown that the resulting algorithm is locally quadratically convergent. A global convergence proof is given for matrices with separated eigenvalues and with relatively small off-diagonal elements. Numerical examples along with comparisons to the QR method are presented.

Linear Algebra and its Applications, 1998

We provide an algorithmic characterization of H-matrices. When A is an H-matrix, this algorithm determines a positive diagonal matrix D such that AD is strictly row diagonally dominant. In effect, D is produced iteratively by quantifying and * Work supported by a Natural Sciences and Engineering Research Council grant.

Applied Mathematics and Computation, 2009

Class of H-matrices plays an important role in various scientific disciplines, in economics, for example. However, this class could be used in order to get various benefits in other linear algebra fields, like determinant estimation, Perron root estimation, eigenvalue localization, improvement of convergence area of relaxation methods, etc. For that reason, it seems important to find a subclass of H-matrices, as wide as possible, and expressed by explicit conditions, involving matrix elements only. One step forward in this direction, starting from Gudkov matrices, from one side, and S-SDD matrices, from the other side, will be presented in this paper.

Applied Mathematics and Computation, 2016

In this paper we start from the known lower bounds for minimal singular value of the matrices possessing certain kind of the diagonal dominance property, and derive Euclidean norm estimates of the inverses of several new subclasses of the block H-matrices. The motivation comes from applications where the matrix in question has distinguished block structure, which can be exploited to obtain useful information. An example arising from ecological modeling illustrates the benefits of the presented approach.

SIAM Journal on Matrix Analysis and Applications, 1990

The inertia of intervals and lines of matrices is investigated. For complex n x n matrices A and B it is shown that. under mild nonsingularity conditions. A + IB changes inertia at no more than n 2 real values of I. Conditions are given for the constancy of the inertia of A + IB, where I lies in a real interval. These conditions generalize and organize some known results. Key words. inertia, constant inertia, inertia change point. interval of matrices. matrix stability, Lyapunov operators, Z-matrices AMS(MOS) subject classification. 15 1. Introduction. Bialas [1], Johnson and Rodman [4], Villiaho [7], and Fu and Barmish [2], have recently studied the inertia of intervals and lines of matrices. We extend these investigations under nonsingularity conditions. While some of our results are not difficult and are related to known results, taken together they show interrelations between various types of conditions, and as such they organize knowledge in this area of inertia theory. Let A and B be square complex matrices and suppose there is a real t such that the Lyapunov matrix L(A + tB) associated with A + tB is nonsingular. We show that A + tB changes inertia at no more than n 2 values of t. Let T be an interval, i.e., a connected subset of the real numbers. Under the assumption that L(A) is nonsingular, we state our principal condition, (CI) A + tB has constant inertia of type (rr, P, 0) for every tin T, and we compare several other conditions (some obviously equivalent) to (CI). Some of these conditions involve the real eigenvalues of A -IB and of L(A)-I L(B). Each of the conditions either implies or is implied by (CI). but not all are equivalent in general. By adding additional requirements on a single matrix or on the interval, such as stability, the reality of all eigenValues, or a condition we call Property X (which Z-matrices satisfy), some implications in one direction become equivalences. Section 2 of our paper contains notation, definitions. and some well-known results stated for easy reference. Section 3 contains preliminary results on eigenvalues and results on changes of inertia. Our main results on intervals with constant inertia, summarized above, may be found in § 4. In § 5 we give some applications to the convex hull of two matrices. We derive results from [I] -[ 4] and .

A necessary and sufficient condition for the stability of n = n matrices with real entries is proved. Applications to asymptotic stability of equilibria for vector fields are considered. The results offer an alternative to the well-known Routh᎐Hurwitz conditions. ᮊ 1998 Academic Press

Esaim: Proceedings, 2007

We describe an algorithm for estimating the H ∞ -norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the H ∞ -norm and then uses extrapolation to improve these estimates.

Linear Algebra and its Applications, 1996

Dedicated to Ciprian Foias on the occasion of his sixtieth birthday. ABSTRA( ;'1" This paper deals with the two-block tI ~ control problem for distributed plants with finitely inanv unstable modes. VCe assume that weighting filters in the H" mixed-sensitivity problem are finite-dimensional. Then tile corresponding optilnal two-blc~.k problem can be solved by finding the Sehmidt pairs of a ttankel operator whose svmbol is of the tbrm m*(m~u +~) where u E Ntt ~. g, EII ~, and m 2 0~' , 11" and ,l I ~ tf ~ are inner; and the suboptimal two-bl(×'k problem can he solved Iw finding the solutions of certain functional equations very similar to the ones satisfied Iw the Schmit pairs of the aNwe-mentioned Hankel operator. In this paper a imified approach is proposed for solving both the optimal and suboptirrud two-blc×.k *

2008

In this book, the methods for localization of eigenvalues of matrices and matrix functions based on constructing and studying the generalized Lyapunov equation are presented. The theory of linear equations and operators in matrix space is developed. The known theorems on the inertia of Hermitian solutions of matrix equations are generalized. New algebraic methods for stability analysis, an evaluation of spectrum and representation of solutions of linear arbitrary order differential and difference systems are worked out. The methods for research and comparison of systems in partially ordered space are developed. The theoretical results are used in the analysis and synthesis problems for dynamical systems.