Generalized products and associated structures on Euclidean spaces

Abstract

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.