A Theory of Matrices of Complex Elements

The articles [11], [14], [1], [4], [2], [15], [6], [10], [9], [3], [8], [7], [13], [12], and [5] provide the terminology and notation for this paper. The following two propositions are true: (1) 1 = 1CF . (2) 0CF = 0. Let A be a matrix over C. The functor ACF yields a matrix over CF and is defined by: (Def. 1) ACF = A. Let A be a matrix over CF. The functor AC yielding a matrix over C is defined by: (Def. 2) AC = A. We now state four propositions: (3) For all matrices A, B over C such that ACF = BCF holds A = B. (4) For all matrices A, B over CF such that AC = BC holds A = B. (5) For every matrix A over C holds A = (ACF)C. (6) For every matrix A over CF holds A = (AC)CF . Let A, B be matrices over C. The functor A + B yielding a matrix over C is defined as follows: