The number of zeros of a tight sign-central matrix

Abstract

A real matrix A is called sign-central if A ∼ x = 0 has a nonzero nonnegative solution x for every matrix A ∼ with the same sign pattern as A. A sign-central matrix A is called tight sign-central if the Hadamard(entrywise) product of any two columns of A contains a negative component. Hwang et al. [S.G. Hwang, I.P. Kim, S.J. Kim, X.D. Zhang, Tight sign-central matrices, Linear Algebra Appl. 371 (2003) 225–240] proved that, for a positive integer m, there exists an m × n (0, 1, −1) tight sign-central matrix A with no zero rows if and only if m + 1 ⩽ n ⩽ 2 m . They also determined the lower bound of the number of columns of a tight sign-central matrix with no zero rows in terms of the number of rows and the number of zero entries of the matrix along with the characterization of the equality case. For an m × n matrix A, the sparsity of A is the ratio σ( A)/ mn where σ( A) denotes the number of zero entries of A. In this paper, we determine the maximum number and the minimum number of zero entries of an m × n tight sign-central matrix with no zero rows for each pair ( m, n) of positive integers with m + 1 ⩽ n ⩽ 2 m . We also determine the maximum sparsity of tight sign-central matrices with m nonzero rows in terms of m for each positive integer m.