Necessary and sufficient conditions for the existence of a positive definite solution for the matrix equation X + +i=1 m A i T Xδi Ai = i

In this paper necessary and sufficient conditions for the matrix equation x + +i=1 m ai t xδi ai = i to have a positive definite solution x are derived,where - 1 < δi < 0,i is an n × n identity matrix,a i are n × n nonsingular real matrices,and m is an odd positive integer. these conditions are used to propose some properties on the matrices a i,i = 1,2,.,m. moreover,relations between the solution x and the matrices a i are derived. © 2013 naglaa m. el-shazly.