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Positive definite solutions of the matrix equations. (English) Zbl 1198.15014
Summary: We investigate nonlinear matrix equations X±A * X-qA=Q where q1. We derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations. We provide a sufficient condition for the equation X+A * X-qA=Q to have two different positive definite solutions and several sufficient conditions for the equation X-A * X-qA=Q to have a unique positive definite solution. We also propose iterative methods for obtaining positive definite solutions of these equations. Numerical examples are given to illustrate the effectiveness of the methods.
MSC:
15A24Matrix equations and identities
15B48Positive matrices and their generalizations; cones of matrices
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