Exact results for perturbation to total positivity and to total nonsingularity

Abstract

A study of the maximum number of equal entries in totally positive and totally nonsingular m-by-n, matrices for small values of m and n, is presented. Equal entries correspond to entries of the totally nonnegative matrix J that are not changed in producing a TP or TNS matrix. It is shown that the maximum number of equal entries in a 7-by-7 totally positive matrix is strictly smaller than that for a 7-by-7 totally non-singular matrix, but, this is the first pair (m; n) for which these maximum numbers differ. Using point-line geometry in the projective plane, a family of values for (m; n) for which these maximum numbers differ is presented. Generalization to the Hadamard core, as well as larger projective planes is also established. Finally, the relationship with C4 free graphs, along with a method for producing symmetric TP matrices with maximal symmetric arrangements of equal entries is discussed.