The diffraction of acoustic, electromagnetic and elastic waves by an edge is a subject of continuing mathematical and physical importance. The physical motivation for research is the vast number of technological and geophysical applications of the fundamental models of linear wave propagation in continuous media, while the mathematical drive is to understand and devise new analytical and numerical techniques for their quantification. This paper is concerned with certain physical aspects of diffraction which have proven to give the most troublesome of mathematical difficulties. These are mode conversion, that is when the incident wave is scattered among different modes supported by the propagating medium, and edge or corner singularities, when the diffracting obstacle creates a `stress singularity'. It is shown, by way of a particular example, that the exact solution of diffraction problems involving these phenomena will in general give rise to a vector Wiener-Hopf equation with a matrix kernel whose factorization is non-commutative. The main result of this paper is a new procedure, motivated by a precise treatment of both the corner singularity and mode conversion, for determining the factors in this case.