Recently a number of writers have discussed interesting developments in the theory of not completely reducible matrix sets and non-semisimple algebras.' Here we have made use of some of these concepts and methods to study matrix algebras over an algebraically closed field. We shall discuss in some detail in this introduction the definitions that are used and the theorems that are developed. Let W denote a matrix algebra, with unit element, over an algebraically closed field K. 2f will be taken in reduced form, by which we shall mean that 91 is exhibited with only zeros above the main diagonal, with irreducible constituents of 2f in the main diagonal, and that 9f is expressible as a direct sum of its radical and a semisimple subalgebra which latter has non-zero components only in the irreducible constituents of V:2