An alternative approach for solving the quadratic matrix equation C∗X∗AXC+C∗X∗B+B∗XC+D=0

Abstract The quadratic matrix equation C ∗ X ∗ A X C + C ∗ X ∗ B + B ∗ X C + D = 0 , where A ∈ ℂ n × n , B ∈ ℂ n × p , C ∈ ℂ m × p and D ∈ ℂ p × p are given complex matrices with A ≥ 0 and D ∗ = D , and X ∈ ℂ n × m is a variable matrix to be determined, was first discussed by Fujioka and Hara (1994). The purpose of this paper is to provide an alternative approach to solve this matrix equation. The necessary and sufficient conditions for the equation to have solutions are presented and the general solution of the equation is provided when the stated conditions are satisfied. In contrast to the method proposed by Fujioka and Hara, the strategy adopted in this paper is from the special case to the general one, and the results seem to be more compact.