Solvability of certain systems of matrix equations related to the Penrose equations

The aim of the paper is to establish equivalent conditions for the solvability of certain systems of matrix equations related to the Penrose equations and present their general solution forms in terms of some inner generalized inverses. A focal role in the considerations is played by the system ADA = A, $ (AD)<^>*=AD $ (AD)& lowast;=AD, and DAB = B, in which $ D\in \mathbb {C}<^>{n\times m} $ D is an element of Cnxm is unknown and $ A\in \mathbb {C}<^>{m\times n} $ A is an element of Cmxn, $ B\in \mathbb {C}<^>{n\times p} $ B is an element of Cnxp are known (or the system ADA = A, $ (DA)<^>*=DA $ (DA)& lowast;=DA, and CAD = C, in which $ A\in \mathbb {C}<^>{m\times n} $ A is an element of Cmxn, $ C\in \mathbb {C}<^>{q\times m} $ C is an element of Cqxm are known). The solvability of the systems obtained by extending the above-mentioned systems by the equation DAD = D are investigated as well. By exploiting the inner generalized inverses which emerge in the solutions obtained, four original partial orderings are specified, and their various properties are identified.