Showing papers in "Electronic Journal of Linear Algebra in 1999"
TL;DR: In this article, a strong stability condition is known which depends only on |G| (= (G T G) 1/2 ≥ 0) and C. If a system with coefficients G0 and C satisfies this condition then all systems with the same C and with a G satisfying |G |≥| G0| are also strongly stable.
Abstract: Here, gyroscopic systems are time-invariant systems for which motions can be char- acterizedby properties of a matrix pencil L(λ )= λ 2 I + λG − C ,w hereG T = −G and C> 0. A strong stability condition is known which depends only on |G| (= (G T G) 1/2 ≥ 0) and C. If a system with coefficients G0 and C satisfies this condition then all systems with the same C andwith a G satisfying |G |≥| G0| are also strongly stable. In order to develop a sense of those variations in G0 which are admissible (preserve strong stability), the class of real skew-symmetric matrices G for which this inequality holds is investigated, and also those G for which |G| = |G0|.
42 citations
TL;DR: For inner products de ned by a symmetric inde nite matrix p;q, canonical forms for real or complex P;q-Hermitian matrices, p; q-skew Hermitian matrix and p; Q-unitary matrices are studied under equivalence transformations which keep the class invariant as mentioned in this paper.
Abstract: For inner products de ned by a symmetric inde nite matrix p;q, canonical forms for real or complex p;q-Hermitian matrices, p;q-skew Hermitian matrices and p;q-unitary matrices are studied under equivalence transformations which keep the class invariant.
33 citations
TL;DR: In this paper, a result of Lewis on the extreme properties of the inner product of two vectors in a Cartan subspace of a semisimple Lie algebra is extended.
Abstract: A result of Lewis on the extreme properties of the inner product of two vectors in a Cartan subspace of a semisimple Lie algebra is extended. The framework used is an Eaton triple which has a reduced triple. Applications are made for determining the minimizers and maximizers of the distance function considered by Chu and Driessel with spectral constraint.
23 citations
TL;DR: Quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2 1 vertices.
Abstract: In this paper, quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2 1 vertices This is accomplished by considering the inverse of a matrix of order k 1 readily obtained from the Laplacian matrix It is shown that the algebraic connectivity is 1=(2 2k + 3) +O(1=2)
23 citations
19 citations
TL;DR: In this article, it was shown that the roots of all scalar polynomials whose coeAEcients correspond to the elements of the convex hull of the joint numerical range of the (m+1)-tuple (A0; A1; : : : ; Am).
Abstract: It is shown that the numerical range, NR[P ( )], of a matrix polynomial P ( ) = Am m + : : :+A1 +A0 consists of the roots of all scalar polynomials whose coeAEcients correspond to the elements of the convex hull of the joint numerical range of the (m+1)-tuple (A0; A1; : : : ; Am). Moreover, the elements of the joint numerical range that give rise to scalar polynomials with a common root belonging to NR[P ( )] form a connected set. The latter fact is used to examine the multiplicity of roots belonging to the intersection of the root zones of NR[P ( )]. Also an approximation scheme for NR[P ( )] is proposed, in terms of numerical ranges of diagonal matrix polynomials.
16 citations
TL;DR: In this article, it was shown that for n 2 there is an n n indecomposable orthogonal matrix with exactly k entries equal to zero if and only if 0 k (n 2).
Abstract: It is shown that for n 2 there is an n n indecomposable orthogonal matrix with exactly k entries equal to zero if and only if 0 k (n 2).
14 citations
TL;DR: In this article, upper and lower bounds on the largest and smallest singular values of a tournament matrix M of order n were obtained for most values of n, and the matrices M for which equality holds were characterized.
Abstract: Upper and lower bounds on both the largest and smallest singular values of a tournament matrix M of order n are obtained. For most values of n, the matrices M for which equality holds are characterized.
11 citations
TL;DR: In this paper, new comparison theorems for weak nonnegative splittings of K-monotone matrices are derived which extend some results of Csordas and Varga.
Abstract: The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors. In this paper new comparison theorems for weak nonnegative splittings of K-monotone matrices are derived which extend some results on regular splittings by Csordas and Varga (1984) for weak nonnegative splittings of the same or di erent types.
9 citations
TL;DR: In this paper, the spectral radius of a strongly connected graph is shown to decrease monotonically with the number of chains replacing the original edges of the graph, and a limiting formula for spectral radius for the expanded graph when the lengths of some chains tend to infinity.
Abstract: We replace certain edges of a directed graph by chains and consider the effect on the spectrum of the graph. We show that the spectral radius decreases monotonically with the expansion and that, for a strongly connected graph that is not a single cycle, the spectral radius decreases strictly monotonically with the expansion. We also give a limiting formula for the spectral radius of the expanded graph when the lengths of some chains replacing the original edges tend to infinity. Our proofs depend on the construction of auxiliary nonnegative matrices of the same size and with the same support as the original adjacency matrix.
8 citations
TL;DR: The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 are investigated in this article, where the authors show that the non-zero elements can be represented by matrices with rank 2.
Abstract: The possible dimensions of spaces of matrices over GF(2) whose nonzero elements all have rank 2 are investigated
TL;DR: In this paper, the spectral condition number (Tn(b)) of sequences of Toeplitz matrices was studied and it was shown that if b is a trigonometric polynomial, then it cannot have strong zeros unless it vanishes identically.
Abstract: The paper deals with the spectral condition numbers (Tn(b)) of sequences of Toeplitz matrices Tn(b) = (bj k) n j;k=1 as n goes to in nity. The function b(ei ) = P k bke ik is referred to as the symbol of the sequence fTn(b)g. It is well known that (Tn(b)) may increase exponentially if the symbol b has very strong zeros on the unit circle T = fz 2 C : jzj = 1g, for example, if b vanishes on some subarc of T. If b is a trigonometric polynomial, in which case the matrices Tn(b) are band matrices, then b cannot have strong zeros unless it vanishes identically. It is shown that the condition numbers (Tn(b)) may nevertheless grow exponentially or even faster to in nity. In particular, it is proved that this always happens if b is a trigonometric polynomial which has no zeros on T but nonzero winding number about the origin. The techniques employed in this paper are also applicable to Toeplitz matrices generated by rational symbols b and to the condition numbers associated with lp norms (1 p 1) instead of the l norm.
TL;DR: A detailed implementation of their algorithm, with some extensions to possibly reducible matrices, is further described in the present paper.
Abstract: I. Bar-On, B. Codenotti, and M. Leoncini presented a linear time algorithm for checking the nonsingularity of general tridiagonal matrices [BIT, 36:206, 1996]. A detailed implementation of their algorithm, with some extensions to possibly reducible matrices, is further described in the present paper.
TL;DR: In this article, a bitangential interpolation problem for upper triangular matrices with HilbertSchmidt norm is studied and the description of all solution in terms of Beurling{Lax representation is given.
Abstract: The space of upper triangular matrices with Hilbert{Schmidt norm can be viewed as a nite dimensional analogue of the Hardy space H2 of the unit disk when one introduces the adequate notion of \point" evaluation. A bitangential interpolation problem in this setting is studied. The description of all solution in terms of Beurling{Lax representation is given.