5 research outputs found

    Positive Definite Solutions of the Nonlinear Matrix Equation X+AHXˉ−1A=IX+A^{\mathrm{H}}\bar{X}^{-1}A=I

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    This paper is concerned with the positive definite solutions to the matrix equation X+AHXˉ−1A=IX+A^{\mathrm{H}}\bar{X}^{-1}A=I where XX is the unknown and AA is a given complex matrix. By introducing and studying a matrix operator on complex matrices, it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation W+BTW−1B=IW+B^{\mathrm{T}}W^{-1}B=I which has been extensively studied in the literature, where BB is a real matrix and is uniquely determined by A.A. It is also shown that if the considered nonlinear matrix equation has a positive definite solution, then it has the maximal and minimal solutions. Bounds of the positive definite solutions are also established in terms of matrix AA. Finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

    Positive definite solution of two kinds of nonlinear matrix equations

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    Abstract. Based on the elegant properties of the Thompson metric, we prove that the following two kinds of nonlinear matrix equations always have a unique positive definite solution. Iterative methods are proposed to compute the unique positive definite solution. We show that the iterative methods are more effective as δ = max{|δi|, i = 1, 2, · · · , m} decreases. Perturbation bounds for the unique positive definite solution are derived in the end

    Positive definite solution of the matrix equation X=Q+A^H(I⊗X-C)^(-δ)A

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