Preservers of term ranks of symmetric matrices

The term rank of an n x n matrix A is the least number of lines (rows or columns) needed to include all the nonzero entries in A. In this paper, we study linear operators that preserve term ranks of n x n symmetric matrices with entries in a commutative antinegative semiring S and with all diagonal entries zero. Consequently, we show that a linear operator T on symmetric matrices with zero diagonal preserves term rank if and only if T preserves term ranks 2 and k (>= 3) if and only if T preserves term ranks 3 and k (>= 4). Other characterizations of term-rank preservers are also given. (C) 2011 Elsevier Inc. All rights reserved.