The nonlinear matrix equation X − A * X q A = Q with 0 < q < 1 is investigated. Two perturbation estimates for the unique positive definite solution of the equation are derived. The theoretical results are illustrated by numerical examples.

In this paper, the nonlinear matrix equation X − A * e X A = I is studied. Some sufficient and necessary conditions for the existence of the Hermitian positive definite solution are given. Then the distribution of the solution is discussed. At last, the basic fixed point iterative method for obtaining the unique positive definite solution is constructed.

We consider the nonlinear matrix equationX=Q+A∗(X^−C)−1A, whereQis positive definite,Cis positive semidefinite, andX^is the block diagonal matrix defined byX^=diag(X,X,…,X). We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

A new inequality on the minimum eigenvalue for the Fan product of nonsingular M-matrices is given. In addition, a new inequality on the spectral radius of the Hadamard product of nonnegative matrices is also obtained. These inequalities can improve considerably some previous results.

By using the fixed point theorem for monotone maps in a normal cone, we prove a uniqueness theorem for the positive definite solution of the matrix equation = + * ( ) , where is a monotone map on the set of positive definite matrices. Then we apply the uniqueness theorem to a special equation = + * (̂− ) and prove that the equation has a unique positive definite solution when̂≥ and > 1and 0 < < 1. For this equation the basic fixed point iteration is discussed. Numerical examples show that the iterative method is feasible and effective.