The range of combined matrices and doubly quasi-stochastic matrices of order 3

Abstract

The combined matrix of a nonsingular matrix A is the matrix C(A)=A∘A−t, where ∘ means the Hadamard (entrywise) product. Since the sum of the entries of each row and each column of C(A) is one, when the combined matrix is a real matrix, it is a doubly quasi-stochastic matrix. In this work, doubly quasi-stochastic combined matrices are studied. The characterization of all real matrices such that their combined matrix is doubly quasi-stochastic of order 3 is obtained, and the study of the class of totally positive matrices is given.