Some inequalities for the nonlinear matrix equation Xs+A∗X-tA=Q: Trace, determinant and eigenvalue

The nonlinear matrix equation Xs+A∗X-tA=Q is investigated, where Q is a Hermitian positive definite matrix. We present the upper bound of det(AA∗)1n when the nonlinear matrix equation Xs+A∗X-tA=Q has Hermitian positive definite solution. The bound of det(AA∗)1n improves the corresponding result in Duan and Liao (2008) [6, Theorem 3.3] and Yin et al. (2009) [17, Theorem 2.1]. We also get the bounds of detX1n and trX for the existence of a Hermitian positive definite solution, which generalize and improve the corresponding conclusions of Zhao (2013) [22]. We obtain some bounds for the eigenvalues of the Hermitian positive definite solution. As compared with earlier works on these topics, the results we present here are more general, and the analysis here is much simpler. Finally, we derive tight bounds about partial sum and partial product about the eigenvalues of the solution X for the nonlinear matrix equation Xs+A∗X-tA=Q. These present improvements and completions for the existing bounds.