Algebraic and geometric structures of matrices under STA and STP

The purpose of this paper is to provide a general algebraic and geometric structure for the set of matrices. First, the identity equivalence is introduced to the set. Then two operations STA and STP are introduced, which are the generalizations of conventional matrix addition and matrix product respectively. Using them, the set of matrices is partitioned into several vector spaces. A topological structure is proposed to make each vector space a second countable, Hausdorff space. Then a differential structure is posed to make each vector space a differential manifold with multi-dimensions. Finally, the multi-dimensional Lie group and Lie algebra structures are obtained for the set of all square matrices. Some applications are also explored.