The paper shows that floating-point matrix operations can be implemented in a way which leads to reasonable mathematical structures as well as to sensible compatibility properties between these structures and the structure of the real matrices. It turns out, for instance, that all the rules of the minus-operator for real matrices can be saved and that for all elements which are comparable with 0 with respect to ≦ and ≧ the same rules for inequalities hold as for real matrices. These structures also occur in other fields of mathematics [5], [6], [7]. They allow many theoretical considerations with floating-point matrices. The proposed implementation, furthermore, leads to a higher accuracy of floating-point matrix operations and allows a much simpler error analysis (Theorem 2.5).