The solutions to the quadratic matrix equation X*AX+B*X+D=0

Abstract The necessary and sufficient conditions for the existence and the explicit expression for the general solution are obtained for the quadratic matrix equation X * A X + B * X + D = 0 , where A ∈ C n × n , B ∈ C n × p and D ∈ C p × p are given complex matrices with D * = D and A * = A , and ( · ) * denotes the conjugate transpose of a complex matrix. Furthermore, a type of inverse statistical problem which arises in satellite meteorology is discussed.