ELA, Volume 9, pp. 93-107, May 2002, abstract.

The Symmetric linear matrix equation

Andre C.M. Ran and Martine C.B. Reurings 

In this paper sufficient conditions are derived
for the existence of unique and positive definite 
solutions of the matrix equations 
     X-A_1^*XA_1- ... -A_m^*XA_m = Q
and 
     X+A_1^*XA_1+ ... +A_m^*XA_m = Q. 
In the case there is a unique solution which is 
positive definite an explicit expression for 
this solution is given.