On real solutions of the equation \(\Phi^t(A)=\frac{1}{n} J_n\) (Q2706299)

Denote by \(M_n({\mathbf R})\), \(GL_n({\mathbf R})\), \(I_n\) and \(J_n\) respectively the set of all \(n\times n\) real matrices, the set of all \(n\times n\) nonsingular matrices, the identity matrix and the all-one matrix. The Hadamard product of the matrices \(A=(a_{ij})\) and \(B=(b_{ij})\) from \(M_n({\mathbf R})\) is defined by \(A\circ B=(a_{ij}b_{ij})\in M_n({\mathbf R})\). Define the map \(\Phi : GL_n({\mathbf R})\rightarrow M_n({\mathbf R})\) by \(\Phi (A)=A\circ A^{-T}\) where \(A^{-T}=(A^{-1})^T\) is the inverse transpose of \(A\). The map \(\Phi\) arises in mathematical control theory in chemical engineering design problems. Every matrix \(\Phi (A)\) has row and column sums \(1\) but the converse is not true. \textit{C. R. Johnson} and \textit{H. M. Shapiro} [SIAM J. Algebraic Discrete Methods 7, No. 4, 627-644 (1986; Zbl 0607.15013)] show that the equation \(\Phi (A)=\frac{1}{3}J_3\) has no real solutions. \textit{X. Zhang, Z. Yang} and \textit{C. Cao} [(*) SIAM J. Matrix Anal. Appl. 21, No. 2, 642-645 (2000; Zbl 0957.15010)] study the existence of real solutions to the equation (1) \(\Phi ^t(A)=\frac{1}{n}J_n\), \(t\in {\mathbf N}^*\) where \(\Phi ^t\) is \(\Phi\) applied \(t\) times. NEWLINENEWLINENEWLINEThe matrices \(A\) and \(B\) are diagonally (resp. permutation) equivalent if there exist non-singular diagonal (resp. permutation) matrices \(D\) and \(E\) such that \(A=DBE\). In the paper new solutions to (1) are obtained by generalizing the method from (*). Some of these solutions are neither diagonally nor permutation equivalent to the ones from (*).