Some investigation on Hermitian positive-definite solutions of a nonlinear matrix equation

Abstract

In this paper, the Hermitian positive-definite solutions of the matrix equation X^s+A*X^−tA=Q are considered. New necessary and sufficient conditions for the equation to have a Hermitian positive-definite solution are derived. In particular, when A is singular, a new estimate of Hermitian positive-definite solutions is obtained. In the end, based on the fixed point theorem, an iterative algorithm for obtaining the positive-definite solutions of the equation with Q=I is discussed. The error estimations are found.

Keywords:

• nonlinear matrix equation
• Hermitian positive-definite solution
• necessary and sufficient condition
• iteration

2000 AMS Subject Classifications:

• 15A24
• 15A48
• 65H05
• 65F10
• 65F05