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Some inequalities for the nonlinear matrix equations. (English) Zbl 1273.15006

Summary: The nonlinear matrix equations \(X \pm A^\ast X^{-1}A = \mathcal Q\) have many applications in control theory, dynamic programming and statistics. In the present work, by using some related matrix equalities and linear algebraic techniques, we propose some trace and determinant inequalities for the solution of the above nonlinear matrix equations. Furthermore, we give some inequalities for the eigenvalues of their solutions.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
15A24 Matrix equations and identities
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