Linear Algebra and its Applications 

About: Linear Algebra and its Applications is an academic journal. The journal publishes majorly in the area(s): Matrix (mathematics) & Eigenvalues and eigenvectors. It has an ISSN identifier of 0024-3795. Over the lifetime, 14461 publication(s) have been published receiving 262237 citation(s).

Completely positive linear maps on complex matrices

Abstract: A linear map Φ from M n to M m is completely positive iff it admits an expression Φ(A)=Σ i V ∗ i AV i where Vi are n×m matrices.

Applications of second-order cone programming

Abstract: In a second-Order cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of second-Order (quadratic) cones. SOCPs are nonlinear convex Problems that include linear and (convex) quadratic programs as special cases, but are less general than semidefinite programs (SDPs). Several efficient primaldual interior-Point methods for SOCP have been developed in the last few years. After reviewing the basic theory of SOCPs, we describe general families of Problems that tan be recast as SOCPs. These include robust linear programming and robust leastsquares Problems, Problems involving sums or maxima of norms, or with convex hyperbolic constraints. We discuss a variety of engineering applications, such as filter design, antenna array weight design, truss design, and grasping forte optimization in robotics. We describe an efficient primaldual interior-Point method for solving SOCPs, which shares many of the features of primaldual interior-Point methods for linear program

Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics

Abstract: A three-way array X (or three-dimensional matrix) is an array of numbers xijk subscripted by three indices. A triad is a multiplicative array, xijk = aibjck. Analogous to the rank and the row rank of a matrix, we define rank (X) to be the minimum number of triads whose sum is X, and dim1(X) to be the dimensionality of the space of matrices generated by the 1-slabs of X. (Rank and dim1 may not be equal.) We prove several lower bounds on rank. For example, a special case of Theorem 1 is that rank(X)⩾dim 1 (UX) + rank(XW) − dim 1 (UXW) , where U and W are matrices; this generalizes a matrix theorem of Frobenius. We define the triple product [A, B, C] of three matrices to be the three-way array whose (i, j, k) element is given by ⩞rairbjrckr; in other words, the triple product is the sum of triads formed from the columns of A, B, and C. We prove several sufficient conditions for the factors of a triple product to be essentially unique. For example (see Theorem 4a), suppose [A, B, C] = [ A , B , C ] , and each of the matrices has R columns. Suppose every set of rank (A) columns of A are independent, and similar conditions hold for B and C. Suppose rank (A) + rank (B) + rank (C) ⩾ 2R + 2. Then there exist diagonal matrices Λ, M, N and a permutation matrix P such that A = APΛ, B = BP M , C = CP N . Our results have applications to arithmetic complexity theory and to statistical models used in three-way multidimensional scaling.

Laplacian matrices of graphs: a survey

Abstract: Let G be a graph on n vertices. Its Laplacian matrix is the n -by- n matrix L ( G ) D ( G )− A ( G ), where A(G) is the familiar (0,1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equivalence, Chemical Applications, and Immanants.

The energy of a graph

Abstract: The energy, E(G), of a simple graph G is defined to be the sum of the absolute values of the eigen values of G. If G is a k-regular graph on n vertices,then E(G)⩽k+ k(n−1)(n−k) =B 2 and this bound is sharp. It is shown that for each ϵ>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k E(G) B 2 . Two graphs with the same number of vertices are equienergetic if they have the same energy. We show that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.