Showing papers in "Linear Algebra and its Applications in 2003"

TL;DR: For almost all graphs the answer to the question in the title is still unknown as mentioned in this paper, and the cases for which the answer is known are surveyed in the survey of cases where the Laplacian matrix is known.

TL;DR: In this paper, it was shown that monic orthogonal polynomials on the unit circle are the characteristic polynomial of certain five-diagonal matrices depending on the Schur parameters.

TL;DR: In this article, it was shown that max-algebra may be regarded as a linear algebraic encoding of a class of combinatorial problems, such as systems of linear equations, eigenvalue-eigenvector problem, linear independence, regularity and characteristic polynomial.

TL;DR: Two procedures for constructing the nearest structured low rank matrix are proposed and are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness.

TL;DR: The singular value distribution (the eigenvalue distribution in the Hermitian case) is studied and characterized for generalized locally Toeplitz sequences in terms of weighted multidimensional Szegő formulas.

TL;DR: An exposition of the spectral theory of normal matrices with quaternion entries is given in this article, where the quaternions are represented by a set of columns and columns.

TL;DR: In this paper, the existence of symmetric positive definite solutions of the two matrix equations X s + A T X − t t A = I n and X s − A T x − t A= I n is proved based on the fixed-point theory.

TL;DR: In this paper, it was shown that the eigenvalues of a convex function defined on an interval I, 0, 0⩽α ⩽1 and A,B n×n complex Hermitian matrices with spectrum in I.

TL;DR: In this article, the usual companion matrix of a polynomial of degree n can be factored into a product of n matrices, n − 1 of them being the identity matrix in which a 2×2 identity submatrix in two consecutive rows (and columns) is replaced by an appropriate 2 × 2 matrix, the remaining being an identity matrix with the last entry replaced by possibly different entry.

TL;DR: In this article, the authors studied the solutions of complex matrix equations X−AXB=C and X −AXB =C, and obtained explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively.

TL;DR: Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix) based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficientMatrices.

TL;DR: In this article, the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti-reflexive with respect to a generalized reflection matrix P solutions of the matrix equation AX = B.

TL;DR: In this article, a simple proof for the inequality that positive definite matrices constitute a Riemannian manifold of negative curvature is given, which leads to generalisations to non-Riemannians and connections with some well-known inequalities of mathematical physics.

TL;DR: For any m × n matrix A, a pivoting strategy based on the idea of local maximum volumes is introduced and a proof of the existence of strong RRLU is given in this paper.

TL;DR: In this article, it was shown that the largest eigenvalue of the Laplacian matrix of a simple graph can be approximated by max 2 d 2 i +d i m ǫ i :1 ⩽i⩽n, where m à i = ∑ j d j −|N i ∩N j |:v i v j ∈E d i.

TL;DR: It is shown that a polynomial eigenvalue problem is well-posed when its eigenvalues are simple and the distance to the nearest ill-posed problem is equal to the reciprocal of the condition number of the eigenvector x.

TL;DR: In this article, spectral decompositions of real circulant matrices are presented, i.e., eigendecompositions and singular value decomposition, for skew right and skew left circulants.

TL;DR: In this paper, it was shown that the conjugacy vector of the group of upper unitriangular matrices of size n×n over Fq is a polynomial in q with integral coefficients and degree [n(n+6)/12.

TL;DR: The present work is a contribution towards the generalization of the most common preconditioners for linear systems whose matrix coefficient is an M-matrix.

TL;DR: It is shown in an explicit form that the last problem contains all classification matrix problems given by quivers or posets, and that every wild classification problem given by a quiver or poset has the same complexity.

TL;DR: In this paper, the main properties of the mapping σ(·,K) were studied, and structural differences existing between the polyhedral case and the nonpolyhedral case were discussed.

TL;DR: In this paper, the authors characterized the largest nullity attained by any positive semi-definite matrix M ∈ M G for t = 1 and 2, for which the maximum nullity is not greater than t.

TL;DR: In this paper, it was shown that U12k and U22k have the largest and second largest spectral radius among the graphs in U+(2k) when k ≥ 3.

TL;DR: In this paper, the authors present sufficient conditions in order that the Leontief price model has nonnegative solutions for the linear system Ax = b, where b⩾ 0.

TL;DR: In this article, the authors were supported by NSF under Grant No.EPS-9874732 and matching support from the State of Kansas and the National Institute of Mental Health.

TL;DR: In this article, the problem of inverse eigenvalue problems of centro-symmetric matrices and their optimal approximation was studied. But the problem was not considered in this paper.

TL;DR: In this article, an upper bound for the Laplacian spectral radius of a tree in terms of the matching number and number of vertices is presented. But the upper bound is not exact.

TL;DR: In this article, it was shown that λ 1(G) is the largest eigenvalue of the adjacency matrix of a graph G and λ 2(G), λ 3 (G) denotes the largest value of the Laplacian matrix of G.

TL;DR: In this paper, a parameterization of the class PD(n) of positive definite n×n matrices using regular vines and partial correlations is presented, and a new approach to various problems involving positive definiteness is suggested.

TL;DR: In this article, the authors considered meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (GCD) matrices, and showed that a semi-multiplicative function can be used to define the determinant of join matrices.