Abstract
For ann x n real matrixX, let ϕ(X)=X ο (X−1)T, where ο stands for the Hadamard (entrywise) product. SupposeA, B, C andD aren x n real nonsingular matrices, and among them there are at least one solutions to the equation ϕ(X)=1/nJn. An equivalent condition which enable\(M = \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right)\) become a real solution to the equation ϕ(X)=1/2nJ2n, is given. As applications, we get new real solutions to the matrix equation ϕ(X)-1/2nJ2n by applying the results of Zhang, Yang and Cao [SIAM. J. Matrix Anal. Appl, 21 (1999), pp: 642–645] and Chen [SIAM. J. Matrix Anal. Appl, 22 (2001), pp:965–970]. At the same time, all solutions of the matrix equation ϕ(X)=1/4J4 are also given.
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