Approximations with Real Linear Modules

Real linear approximation theory is developed further by regarding ? ⁿ as a module over the ring of circlets. By introducing a concept of orthogonality together with the respective Gram–Schmidt orthogonalization process, improved approximations upon the standard complex Hilbert space techniques follow. Related hierarchical bases are devised leading to a new family of rapidly constructible family of unitary matrices. With circlets, so-called oplets are introduced for approximation to improve the singular value decomposition of real matrices. Complex matrix approximation is also considered through finding the nearest real matrix in small rank perturbations.