Positive definite solutions of the nonlinear matrix equation \(X+A^*X^qA=Q\) (\(q>0\)). (Q547990)

The paper deals with positive definite solutions of the nonlinear matrix equation \[ X+A^*X^qA=Q, \;\;q>0, \tag{1} \] where \(Q\) is an \(n \times n\) Hermitian positive definite matrix, \(A\) is an \(n \times n\) nonsingular complex matrix and \(A^*\) is the conjugate transpose of matrix \(A\). In this work, the authors obtain some new properties of the positive definite solutions for the matrix equation (1) and present necessary and sufficient conditions for the existence of Hermitian positive definite solutions of this matrix equation. In addition, they present two iterative methods for obtaining the positive definite solutions for matrix equation (1). Finally, numerical examples confirm the theoretical results and illustrate the effectiveness of the iterative methods.