Matrix Decompositions And Generalized Sylvester Matrix Equations Over Division Rings

In this paper, we consider the simultaneous decompositions for some sets of matrices over an arbitrary division ring F and their applications in generalized Sylvester matrix equations.First, we give all the dimensions of identity matrices in the equivalence canonical form of four matrices over an arbitrary division ring F with compatible sizes:A∈F^m×p, B∈F^m×p, C∈F^s×p, and D∈F^t×p. As applications, we derive some necessary and sufficient conditions for the solvability to some well known systems of matrix equations over an arbitrary division ring using rank conditions.Then, we construct a simultaneous decomposition for a set of seven general matrices over an arbitrary division ring F with compatible sizes: A∈F^m×n, B∈F^m×p1, C∈F^m×p2, D∈F^m×p3, E∈F^q1×n, F∈F^q2×n, and G∈F^q3×n. As applications of the simultaneous matrix decomposition, we give some solvability conditions, general solutions, as well as the range of ranks of the general solutions to some generalized Sylvester matrix equations over an arbitrary division ring F.Finally, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices over an arbitrary division ring with an involutive antiautomorphism with compatible sizes: A_i∈F^pi×ti, B_i∈F^si×qi, C_i∈F^pi×ti+1, D_i∈F^si+1×qi, and E_i∈F^pi×qi,（i = 1,2,3）. We show that from this simultaneous decomposition we can derive some solvability conditions,general solutions, as well as the range of ranks of the general solutions to the system of two-sided coupled generalized Sylvester matrix equations over an arbitrary division ring with an involutive antiautomorphism.Moreover, we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two-sided coupled generalized Sylvester matrix equations with k + 1 unknowns over an arbitrary division ring with an involutive antiautomorphism.