Matrix Equation X ~~ S, ¡À A, ~ * X ~ (-t) = P Hermitian Positive Definite Solution

In this thesis we study the Hermite positive definite solutions of matrix equations X^S + A^*X^-tA = P and X^S - A^*X^-tA = P. For the matrix equations X^S+A^*X^-tA = P, we present some necessary conditions and sufficient conditions for existence of the Hermite positive definite solutions, and obtain intervals of minimum and of maximal eigenvalues of the Hermite positive definite solutions. For the matrix equations X^S - A^*X^-tA = P, A∈ C^n×n and P ∈ H^n×n is proved that the equations have a Hermite positive definite solution, moreover a sufficient condition for uniqueness of a unique Hermite positive definite solution is given.