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 the same sign pattern as B-n is nonsingular. A given SNS matrix determines an equivalence class (with respect to transposition and multiplication by permutation and signature matrices) of SNS matrices, all of which have the same number of zero entries. Such a matrix is maximal if no zero entry can be set nonzero so that the resulting matrix is SNS, and is fully indecomposable if it does not have an (n - k)-by-k zero submatrix 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 the unique equivalence class with the minimum number of zero entries, namely [GRAPHICS] We prove that for n greater than or equal to 5, there is exactly one equivalence class of fully indecomposable maximal SNS matrices with [GRAPHICS] + 1 zero entries. Similarly, for n greater than or equal to 5, we prove that there are exactly two such equivalence classes having [GRAPHICS] + 2 zero entries. For these proofs, we identify two new infinite classes of fully indecomposable maximal SNS matrices, which can be obtained by stretching known SNS matrices.