Nonnegative Combined Matrices

Abstract

The combined matrix of a nonsingular real matrix is the Hadamard (entrywise) product ∘ ( −1 ) . It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of . In this case the combined matrix is doubly stochastic.

Combined matrices of almost strictly sign regular matrices

“…Taking into account Lemma 8 (ii) of [5], we can conclude that the combined matrix of a triangular matrix is the identity matrix.…”

“…In recent years, several studies about combined matrices have been carried out (see [5][6][7][8]). In particular, in [5,6], the authors study some kinds of matrices such that their combined matrices are nonnegative, analyzing their relationship with the zero pattern of A. In this case, that is, when C (A) is nonnegative, the combined matrix is doubly stochastic.…”

Combined matrices of sign regular matrices

“…We have proven that the only nonnegative combined matrix of a TNN matrix is the identity matrix (see Theorem 2.1 of [2]). …”

“…Applications ✩ Research supported by Spanish DGI grant number MTM2010-18674. of the combined matrix can be found in [7] and [9]. In [2], we have studied when the combined matrix of some classes of matrices is nonnegative. More precisely, we have studied the nonnegativity of the combined matrix of totally positive (nonnegative) matrices and totally negative (nonpositive) matrices.…”

Diagonal entries of the combined matrix of a totally negative matrix

“…Let us recall the basic definitions and properties. Some recent works on combined matrices are given in [3] for positive and negative matrices, in [4] for sign regular matrices and in [5] for H-matrices. Test to recognize total positive matrices has been studied in [15].…”