A matrix whose entries consist of elements from the set is a sign pattern matrix. Using a linear algebra theoretical approach we generalize of some recent results due to Hall, Li and others involving the inertia of symmetric tridiagonal sign matrices.
A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary...
We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix and of mₙ rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mₙ/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges...
We define the linear capacity of an algebraic cone, give basic properties of the notion and new formulations of certain known results of the Matrix Theory. We derive in an explicit way the formula for the linear capacity of an irreducible component of the zero cone of a quadratic form over an algebraically closed field. We also give a formula for the linear capacity of the cone over the conjugacy class of a “generic” non-nilpotent matrix.
We show that a central linear mapping of a projectively embedded Euclidean -space onto a projectively embedded Euclidean -space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity . This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.
Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) with d ∈ N for nonsingular , i=1,...,n.