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In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I  and  X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C ,  and  D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

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