Algebraic Characterizations of Relationships between Different Linear Matrix Functions

Let $f(X_1,X_2,\dots ,X_k)$ be a matrix function over the field of complex numbers, where $X_1,X_2,\dots ,X_k$ are a family of matrices with variable entries. The purpose of this paper is to propose and investigate the relationships between certain linear matrix functions that regularly appear in matrix theory and its applications. We shall derive a series of meaningful, necessary, and sufficient conditions for the collections of values of two given matrix functions to be equal through the cogent use of some highly selective formulas and facts regarding ranks, ranges, and generalized inverses of block matrix operations. As applications, we discuss some concrete topics concerning the algebraic connections between general solutions of a given linear matrix equation and its reduced equations.