Fuzzified Matrix Space and Solvability of Matrix Equations

A fuzzified matrix space consists of a collection of matrices with a fuzzy structure, modeling the cases of uncertainty on the part of values of different matrices, including the uncertainty of the very existence of matrices with the given values. The fuzzified matrix space also serves as a test for the admissibility of certain approximate solutions to matrix equations, as well as a test for the approximate validity of certain laws. We introduce quotient structures derived from the original fuzzified matrix space and demonstrate the transferability of certain fuzzy properties from the fuzzified matrix space to its associated quotient structures. These properties encompass various aspects, including the solvability and unique solvability of equations of a specific type, the (unique) solvability of individual equations, as well as the validity of identities such as associativity. While the solvability and unique solvability of a single equation in a matrix space are equivalent to the solvability and unique solvability in a certain quotient structure, we proved that the (unique) solvability of a whole type of equations, as well as the validity of a certain algebraic law, are equivalent to the (unique) solvability and validity in all the quotient structures. Consequently, these quotient structures serve as an effective tool for evaluating whether specific properties hold within a given fuzzified matrix space.