Solvability of certain systems of matrix equations related to the Penrose equations
Dijana Mosić, Oskar Maria Baksalary · Linear and Multilinear Algebra · 2025
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, and DAB = B, in which D∈Cn×m is unknown and A∈Cm×n, B∈Cn×p are known (or the system ADA = A, (DA)∗=DA, and CAD = C, in which A∈Cm×n, C∈Cq×m 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.