About Hermitian Positive Definite Solutions of a Type of Nonlinear Matrix Equations

Let A be an n × n nonsingular matrix. In image processing, we must solve a system of linear equations [B. L. Buzbee, G. H. Golub, C. W. Nielson (1970)] Mx = f, however, the solving of the System Mx = f can be transformed to the solving of the equations X + A* X
^-q
A = I. In this paper, we study the Hermitian positive definite solutions of the matrix equation X + A* X
^-q
A = I . We give an equivalent equation of X + A*X
^-q
A = I when the matrix equation has a Hermitian positive definite solution.