On equivalence of matrices (Q2000881)

The two mostly used operations on the set of matrices are the usual matrix product and matrix addition. However, these operations are restricted by the matrix dimension. In the present article, the author develops a new matrix theory, which overcomes the dimension barrier by extending the matrix product and matrix addition to two matrices which do not meet the classical dimension requirement. The approach is consistent with the classical definitions. Many related concepts are extended. For instance, the characteristic functions, the eigenvalues and eigenvectors of a square matrix can be extended to certain non-square matrices. Also, the Lie algebraic structure can be extended to dimension-varying square matrices. Though the extended operations are applicable to certain sets of matrices with different dimensions, such matrices fail to span vector spaces. To overcome this obstacle, the author introduces certain equivalence relations. Then, the quotient spaces, called the equivalence spaces, become vector spaces. Two equivalence relations are proposed. They are matrix equivalence (M-equivalence) and vector equivalence (V-equivalence). Then, many nice algebraic, analytic, and geometric structures are developed on the M-equivalence spaces.