Showing papers in "Linear Algebra and its Applications in 2007"

TL;DR: In this article, the authors survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix for regular graphs.

TL;DR: A survey of algebraic connectivity of a graph G is given in this paper, where the second smallest eigenvalue of the Laplacian of the graph G, denoted a(G), is considered.

TL;DR: A novel way of constructing generalized state space representations [E, A, B, C, D] of interpolants matching tangential interpolation data by using the Loewner and shifted Loewer matrices.

TL;DR: This article uses field theory to provide detailed conditions on real and complex ETFs and describes restrictions on harmonic ETFs, a specific type of complex ETF that appears in applications.

TL;DR: The current state of knowledge on the problem of determining the minimum rank of a graph and related issues is surveyed.

TL;DR: It is proved that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is O ( n ) for bounded-degree planar graphs and two-dimensional meshes and O(n1/d) for well-shaped d- dimensional meshes.

TL;DR: In this article, the authors provide an accessible and intuitive proof of Kruskal's condition for the complex-valued CP decomposition, which can be adapted for the CP decompositions of real-valued three-way arrays.

TL;DR: In this paper, lower and upper bounds for the Estrada index of G are established in terms of the number of vertices and number of edges of the graph and some inequalities between EE and the energy of G were obtained.

TL;DR: In this article, the spectral radius of a Turan graph of order n was shown to be at most 2 m/n > 1 /( 2 m + 2 n ) unless G = T r ( n ).

TL;DR: An algorithm is presented and analyzed that, when given as input a d-mode tensor A, computes an approximation A by performing the following for each of the d modes: first, form (implicitly) a matrix by “unfolding” the tensor along that mode; then, choose columns from the matrices thus generated; and finally, project the TensorAlong that mode onto the span of those columns.

TL;DR: In this paper, the Moore-Penrose inverse (MP-inverse) was studied in the setting of rings with involution and the relation between regular, MP-invertible and well-supported elements was analyzed.

TL;DR: This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors and gives new proofs of existing results suitably generalizing, restating and refining them.

TL;DR: In this article, it was shown that any point of a compact max-plus convex subset of (R ∪ { - ∞ } ) n can be written as the max plus convex combination of at most n ǫ+ǫ 1 of the extreme points of this subset.

TL;DR: In this paper, the performance of the combination technique is analyzed using a projection framework and the C/S decomposition, and modified combination coefficients are derived which are optimal in a certain sense.

TL;DR: This work considers a matrix approximation problem arising in the study of entanglement in quantum physics, and discusses this approximation problem for a composite system with two subsystems and shows that it can be written as a convex optimization problem with special structure.

TL;DR: In this paper, it was shown that if a tree G is a star-like tree, then it is determined by its Laplacian spectrum, and that trees with the same adjacency spectrum as a starlike tree can also be classified as star trees.

TL;DR: Gutman and Vidovic as discussed by the authors showed that P n 6, 6 is the graph with maximal energy in B n, which gave a partial solution to Gutman's conjecture in 2001.

TL;DR: In this paper, the eigenvalues function of a symmetric real-valued function is studied in the context of Euclidean Jordan algebras, and a formula for the Jacobian of these functions is given.

TL;DR: In this article, the Leonard pair on V is defined as the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix corresponding to X is tridimensional, provided the dimension of V is at least 3.

TL;DR: In this paper, the authors consider the local Cheeger constant which is the minimum Cheeger ratio over all subsets of a specified subset S of vertices in a graph G and consider local cuts that separate a subset of S from G. The proofs are based on the methods of establishing isoperimetric inequalities using random walks and the spectral methods for eigenvalues with Dirichlet boundary conditions.

TL;DR: In this paper, the authors present the Gersgorin type theorems for the left and right eigenvalues of square quaternionic matrices, and conclude the paper with examples showing and summarizing some differences between complex matrices and quaternion matrices.

TL;DR: For a linear relation in a linear space the concepts of ascent, descent, nullity, and defect are introduced and studied in this article, and it is shown that the results of A.E. Taylor and M.A. Kaashoek concerning the relationship between ascent and descent for linear operators remain valid in the context of linear relations, sometimes under the additional condition that the linear relation does not have any nontrivial singular chains.

TL;DR: The algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of Neville elimination to solve a Bernstein–Vandermonde linear system.

TL;DR: In this paper, an observable canonical form is proposed for linear positive systems with respect to an equivalent relation defined by permutations of the state and of the output set, and an algorithm for the construction of a linear observer is stated.

TL;DR: In this paper, Ando and Zhan proved a subadditivity inequality for concave functions for all symmetric norms (in particular for all Schatten p-norms).

TL;DR: In this paper, the authors classify the modular Leonard triples up to isomorphism, up to the antiautomorphism of End(V ) which fixes B and swaps the other two operators.

TL;DR: In this paper, the Lagrange multiplier method is used to find the nearest low-rank correlation matrix, which is then applied to the problem of finding the correlation matrix as part of the calibration of multi-factor interest rate market models to correlation.

TL;DR: In this paper, the existence of equiangular tight frames having n = 2 d − 1 elements drawn from either C d or C d - 1 whenever n is either 2 k - 1 for k ∈ N, or a power of a prime such that n ≡ 3 mod 4.

TL;DR: It is shown that of all graphs of order n with matching number β, the graphs with maximal spectral radius are Kn if n = 2β or 2β + 1; K 2 β + 1 ∪ K n - 2 β - 1 ¯ if 2β-+ 2 ⩽ n K β ⋁ K n- β ¯ or K 2β + 1 ⩽ if n-=-3β-2, where K t

TL;DR: A graph is said to be determined by the adjacency spectrum (DS for short) if there is no other nonisomorphic graph with the same spectrum as mentioned in this paper, and all connected graphs with index at most 2 + 5 are known.