Abstract
Let S be an antinegative semiring. The rank of an m × n matrix B over S is the minimal integer r such that B is a product of an m × r matrix and an r × n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form b i , j b i , l b k , j b k , l with nonzero entries. We know that the isolation number of B is not greater than the rank of it. Thus, we investigate the upper bound of the rank of B and the rank of its support for the given matrix B with isolation number h over antinegative semirings.
Highlights
Upper Bounds for the Isolation Number of a Matrix over SemiringsWe say that a2matrix B = [bi,j ] dominates a matrix D = [di,j ] if bi,j = 0 implies di,j = 0
Introduction andPreliminaries set of the indices of the nonzero columns of Bj, i, j = 1, · · ·, h
We investigate the question: given a fixed h, if the isolation number of B is h what can be the upper bound of the rank of B and the rank of support of B? Some terms not defined here can be found in [5] or in [6]
Summary
We say that a2matrix B = [bi,j ] dominates a matrix D = [di,j ] if bi,j = 0 implies di,j = 0. * B,Correspondence: rS ( B) = k and B = CD is a factorization of B ∈ Mm,n (S), B = c(1) d(1) + c(2) d(2) + · · · + c(k) d(k). (i ) is a matrix of rank 1, the rank of B, rS ( B ), is the minimum number of rank-1 matrices whose sum is B
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