Positive Definite Solution Of The Matrix Equation X+A～*X～(-n)A＝I

This thesis is a research on the the definite solution of the matrix equation X + A*X^-nA = I.In the first part ,we study the monotone ,maximal value and minimum of the function f(x) = xⁿ(1 - x) with x ∈ [0,1] at first. Then under the hypothesis ofp(A*A) ≤ eigenvalue's range of the definite solution of thematrix equation X + A*X^-nA = I.In the second part, we give a sufficient condition of the maximal definite solution of the matrix equation X + A*X^-nA = I on the base of the first part.