Some matrix equations with applications

We establish necessary and sufficient conditions for the solvability to the matrix equation A(1)X(1) + X2B1 + C3X3D3 + C4X4D4 = E-1 (1) and present an expression of the general solution to (1) when it is solvable. As applications, we discuss the consistence of the matrix equation A(1)X + (A(1)X)* + B1YC1 + (B1YC1)* = E1, (2) where * means conjugate transpose, and provide an explicit expression of the general solution to (2). We also study the extremal ranks of X-3 and X-4 and extremal inertias of X-3 + X-3(*) and X-4 + X-4(*) in (1). In addition, we obtain necessary and sufficient conditions for the classical matrix equation AY(3)B + CY4D = E to have Re-nonnegative definite, Re-nonpositive definite, Re-positive definite and Re-negative definite solutions. The findings of this article extend related known results.