Solving the nonlinear matrix equation via a contraction principle

In this paper we consider the nonlinear matrix equation where Q is positive (resp. semidefinite) definite and Mi’s are arbitrary (resp. nonsingular) matrices. We prove that if δ≔max{|δi|:1⩽i⩽m}<1, then the equation has a unique positive definite solution which is realized as the unique fixed point of a strict contraction with the Lipschitz constant less than or equal to δ. Furthermore, we show that the solution map varying over the determining coefficient matrices is continuous.