CLASSES OF SIGN NONSINGULAR MATRICES WITH A SPECIFIED NUMBER OF ZERO ENTRIES

An n-by-n matrix B-n is sign nonsingular (SNS) if every matrix with th e same sign pattern as B-n is nonsingular. A given SNS matrix determin es an equivalence class (with respect to transposition and multiplicat ion by permutation and signature matrices) of SNS matrices, all of whi ch have the same number of zero entries. Such a matrix is maximal if n o zero entry can be set nonzero so that the resulting matrix is SNS, a nd is fully indecomposable if it does not have an (n - k)-by-k zero su bmatrix for some k, where 1 less than or equal to k less than or equal to n - 1. For fixed n, the Hessenberg matrix is known to represent th e unique equivalence class with the minimum number of zero entries, na mely [GRAPHICS] We prove that for n greater than or equal to 5, there is exactly one equivalence class of fully indecomposable maximal SNS m atrices with [GRAPHICS] + 1 zero entries. Similarly, for n greater tha n or equal to 5, we prove that there are exactly two such equivalence classes having [GRAPHICS] + 2 zero entries. For these proofs, we ident ify two new infinite classes of fully indecomposable maximal SNS matri ces, which can be obtained by stretching known SNS matrices.