Diagonal entries of the combined matrix of a totally negative matrix

Abstract

The combined matrix of a nonsingular matrix A is the Hadamard (entrywise) product A • A −1 T . This paper deals with the characterization of the diagonal entries of a combined matrix C(A) of a given nonsingular real matrix A. A partial answer describing the diagonal entries of C(A) in the positive definite case was given by Fiedler in 1964. Recently in 2011, Fiedler and Markham characterized the sequence of diagonal entries of the combined matrix C(A) for any totally positive matrix A when the size is 3. For this case, we characterize totally negative matrices and we find necessary and sufficient conditions for the sequence of diagonal entries of C(A), in both cases, symmetric and nonsymmetric.

Diagonal entries of the combined matrix of sign regular matrices of order three

“…The first case (i) was obtained by Fiedler and Markham [7] and corresponds to the totally positive matrices. The second case (ii) was studied by Bru et al [4] when the totally negative matrices were considered.…”

“…Later Fiedler and Markham in [7] give results for totally positive matrices. Bru et al in [4] study that problem for totally negative matrices. It seems that this problem is not trivial since in all above mentioned papers the results are given for matrices of order three.…”