Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X+∑i=1mAiTXδiAi=I

الملخص EN

In this paper necessary and sufficient conditions for the matrix equation X+∑i=1mAiTXδiAi=I to have a positive definite solution X are derived, where -1<δi<0, I is an n×n identity matrix, Ai are n×n nonsingular real matrices, and m is an odd positive integer.

These conditions are used to propose some properties on the matrices Ai, i=1,2,…,m.

Moreover, relations between the solution X and the matrices Ai are derived.