Necessary and sufficient conditions for the existence of a positive definite solution for the matrix equation \(X + \sum^m_{i = 1} A^T_i X^{\delta_i} A_i = I\) (Q364427)

Summary: In this paper necessary and sufficient conditions for the matrix equation \(X + \sum^m_{i = 1} A^T_i X^{\delta_i} A_i = I\) to have a positive definite solution \(X\) are derived, where \(-1 < \delta_i < 0\), \(I\) is an \(n \times n\) identity matrix, \(A_i\) are \(n \times n\) nonsingular real matrices, and \(m\) is an odd positive integer. These conditions are used to propose some properties on the matrices \(A_i\), \(i = 1, 2, \dots, m\). Moreover, relations between the solution \(X\) and the matrices \(A_i\) are derived.