Relations between matrix sets generated from linear matrix expressions and their applications

Let A+BXCA+BXC and A+BX+YCA+BX+YC be two linear matrix expressions, and denote by {A+BXC}{A+BXC} and {A+BX+YC}{A+BX+YC} the collections of the two matrix expressions when XX and YY run over the corresponding matrix spaces. In this paper, we study relationships between the two matrix sets {A1+B1X1C1}{A1+B1X1C1} and {A2+B2X2C2}{A2+B2X2C2}, as well as the two sets {A1+B1X1+Y1C1}{A1+B1X1+Y1C1} and {A2+B2X2+Y2C2}{A2+B2X2+Y2C2}, by using some rank formulas for matrices. In particular, we give necessary and sufficient conditions for the two matrix set inclusions {A1+B1X1C1}⊆{A2+B2X2C2}{A1+B1X1C1}⊆{A2+B2X2C2} and {A1+B1X1+Y1C1}⊆{A2+B2X2+Y2C2}{A1+B1X1+Y1C1}⊆{A2+B2X2+Y2C2} to hold. We also use the results obtained to characterize relations of solutions of some linear matrix equations.