Balancing Matrices with Line Shifts II

Abstract

Let an arbitrary matrix A = (a _ij), 1 ≤ i ≤ K, 1 ≤ j ≤ L be given with all |a _ij| ≤ 1. By a row shift we mean the act of replacing, for a particular i, all coefficients a _ij in the i-th row by their negatives (-a _ij). A column shift is defined similarly. A line shift denotes either a row or a column shift. Consider the following solitaire game. The player applies a succession of line shifts to A. His object is to make the absolute value of the sum of all the coefficients of A (which we shall denote by |A|) as small as possible. We shall show (answering a question of J. Komlós) that the player can always make |A| ≤ c ₀ where c ₀ is an absolute constant — i.e., independent of K, L, and the initial matrix. We make no attempt to find the minimal possible c ₀.