Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X + Σi=1m AiT Xδi Ai = I

In this paper necessary and sufficient conditions for the matrix equation X + Sigma(m)(i=1) A(i)(T) X-delta i A(i) = I to have a positive definite solution X are derived, where -1 < delta(i) < 0, I is an n x n identity matrix, A(i) are n x n nonsingular real matrices, and m is an odd positive integer. These conditions are used to propose some properties on the matrices A(i), i = 1, 2, ... , m Moreover, relations between the solution.. and the matrices A(i) are derived.