Generalized products and associated structures on Euclidean spaces

We define a generalized product of vectors in an arbitrary finite dimensional, inner product space which depends on a finite number of real parameters and which includes as special cases the usual cross-product in R 3 and the product of n— 1 vectors in Rn. The concepts developed in this part are used to define a generalized determinant function of an arbitrary mx nmatrix. This determinant function allows us to define a general notion of non-singularity for a rectangular matrix which turns out to be necessary and sufficient for the existence of certain one-sided inverses. We obtain families of one-sided inverses of rectangular matrices and use them to construct reflexive generalized inverses which include as a special case the well-known Moore-Penrose inverse.