Central configurations have been of great interest over many years, with the earliest examples due to Euler and Lagrange. There are numerous results in the literature demonstrating the existence of central configurations with specific symmetry properties, using slightly different techniques in each. The aim here is to describe a uniform approach by adapting to the symmetric case the well-known ...

We propose an eigenvalue contractor for symmetric matrices. Given a symmetric interval matrix A and an interval approximation of its eigenvalue sets λ1, . . . ,λn the contractor reduces the entries of A S such that no matrix with eigenvalues in λ1, . . . ,λn is omitted. Our contractor is based on sequentially reducing the entries of A. We discuss properties of the method and demonstrate its per...

We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skewsymmetric matrix pencil orbits and bundles. In other words, we investigate when the closure of the congruence orbit (or bundle) of a skew-symmetric matrix pencil contains the congruence orbit (or bundle) of another ske...

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang (2001) showed that the inertia of a signnonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such matrices. The algorithm runs in O(nm) time ...

A standard way of treating the polynomial eigenvalue problem P (λ)x = 0 is to convert it into an equivalent matrix pencil—a process known as linearization. Two vector spaces of pencils L1(P ) and L2(P ), and their intersection DL(P ), have recently been defined and studied by Mackey, Mackey, Mehl, and Mehrmann. The aim of our work is to gain new insight into these spaces and the extent to which...

In this paper sufficient conditions are derived for the existence of unique and positive definite solutions of the matrix equations X−A1XA1− . . .−A∗mXAm = Q and X+A1XA1+ . . .+ A∗mXAm = Q. In the case there is a unique solution which is positive definite an explicit expression for this solution is given.

In this paper, we present an iterative algorithm for solving the following coupled Sylvester-transpose matrix equations q ∑ j=1 ( AijXjBij + CijX j Dij ) = Fi, i = 1, 2, . . . , p, over the generalized centro-symmetric matrix group (X1, X2, . . . , Xq). The solvability of the problem can be determined by the proposed algorithm, automatically. If the coupled Sylvester-transpose matrix equations ...

Introduced the notion of symmetric circulant matrix on skew field, an easy method is given to determine the inverse of symmetric circulant matrix on skew field , with this method , we derived the formula of determine inverse about several special type of symmetric circulant matrix. Mathematics Subject Classification: 05A19, 15A15

We study Wigner ensembles of symmetric random matrices A = (a ij) i; j = 1; : : : ; n with matrix elements a ij ; i j being independent symmetrically distributed random variables a ji = ij n 1 2 : We assume that Var ij = 1 4 , for i < j, Var ii 6const and that all higher moments of ij also exist and grow not faster than the Gaussian ones. Under formulated conditions we prove the central limit t...