Existence and Uniqueness of the Positive Definite Solution for the Matrix EquationX=Q+A∗(X^−C)

Abstract

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.

Positive Definite Solutions of the Matrix EquationXr-∑i=1mA

“…Duan et al [14] used the Thompson metric to prove that the matrix equation − ∑ =1 * −1 = always has a unique positive definite solution and they gave a precise perturbation bound for the unique positive definite solution. Similar kinds of equations have been investigated by many authors [15][16][17][18][19][20].…”

On Hermitian positive definite solutions of the nonlinear matrix equation $$X-A^{*}e^{X}A=I$$ X - A ∗ e X A = I

“…Gao where F is a map on the set of the n × n positive definite matrices. Most authors have studied this equation with the particular map, such as F(X ) = ±X −1 , ±X ±n , ±X ±q , −( X − C) −1 and so on(see [1][2][3][4][5][6][7][8]). But the study is limited to the case that F(X ) is the power function.…”

A Fixed Point Theorem for Monotone Maps and Its Applications to Nonlinear Matrix Equations

“…This equation was first studied in [12], and a perturbation analysis was discussed in [13]. Recently, in [14], the author provided a new proof for the uniqueness of the positive definite solution of this equation using a change of variable and a fixed point theorem, which is an easier argument than the one used in [12].…”