Ela Possible Numbers of Nonzero Entries in a Matrix with a given Term Rank

Possible numbers of nonzero entries in a matrix with a given term rank

Ela the Moore - Penrose Inverse of a Free Matrix

A matrix is free, or generic, if its nonzero entries are algebraically independent. Necessary and sufficient combinatorial conditions are presented for a complex free matrix to have a free Moore-Penrose inverse. These conditions extend previously known results for square, nonsingular free matrices. The result used to prove this characterization relates the combinatorial structure of a free matr...

Preservers of term ranks and star cover numbers of symmetric matrices

Let Sn(S) denote the set of symmetric matrices over some semiring, S. A line of A ∈ Sn(S) is a row or a column of A. A star of A is the submatrix of A consisting of a row and the corresponding column of A. The term rank of A is the minimum number of lines that contain all the nonzero entries of A. The star cover number is the minimum number of stars that contain all the nonzero entries of A. Th...

Ela the Inverse Eigenvalue and Inertia Problems for Minimum Rank Two Graphs∗

Abstract. Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(G) denote the minimum rank of all matrices in S(G), and mr+(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G with mr(G) = 2 and mr+(G) = k are chara...

Ela Diagonal Entry Restrictions in Minimum Rank Matrices

Let F be a field, let G be a simple graph on n vertices, and let S (G) be the class of all F -valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class, M R (G) consisting of all matrices A ∈ S (G) with rankA = mr (G), the minimum rank among all matrices in S (G)....