On the Hermitian positive definite solution of the nonlinear matrix equation (Q988594)

The matrix equation \(X+\sum_{i=1}^mA_i^*X^{-1}A_i=I\) is considered, where \(A_i\in\mathbb{C}^{n\times n}, i=1,2,\dots,m\). Some necessary and/or sufficient conditions are given for the existence of a Hermitian positive definite solution. For instance, Theorem 2.2 states that such solution exists if and only if all the matrices in the sequence \(\{\phi^j(I)\}_{j=0}^{\infty}\) are positive definite and the sequence \(\{\phi^j(I)^{-1}\}_{j=0}^{\infty}\) is uniformly bounded, where \(\phi(X)=I-\sum_{i=1}^mA_i^*X^{-1}A_i\), \(\phi^1=\phi\) and \(\phi^{j+1}=\phi(\phi^j)\). Two iterative algorithms are provided to determine the maximal positive definite solution. Two examples illustrate the behaviour of the proposed algorithms.