The central goal of our software theory is to solve the composition problem of a software system of any size from sub-systems down to indivisible components either produced in house, or purchased from other manufacturers. The theory guides the gradual or agile design of the system by means of quantitative criteria for the current design quality, and by highlighting problematic coupling spots to be resolved in order to improve the software design. The theory is based upon solid results of linear algebra, and the broadly accepted wisdom that modularity is essential for understandable and maintainable software design. The main algebraic structure of the theory is the Modularity Matrix. It links structors – a generalization of classes – in the matrix columns, to their provided functionals – a generalization of methods – in the matrix rows. The theory predicts that neat modules for any given system are the blocks of a block-diagonal standard matrix.