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On the existence of Hermitian positive definite solutions of the matrix equation X s +A * X -t A=Q. (English) Zbl 1143.15011
Summary: The existence of Hermitian positive definite solutions of the general nonlinear matrix equation X s +A * X -t A=Q is studied systematically and deeply. A new estimate of Hermitian positive definite solutions is derived. Based on a fixed point theorem, some new sufficient conditions and new necessary conditions for the existence of Hermitian positive definite solutions are obtained. In the end, a necessary and sufficient condition for the existence of a Hermitian positive definite solution is proved.
MSC:
15A24Matrix equations and identities
15A42Inequalities involving eigenvalues and eigenvectors
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