GRASSMANN MATRICES, DETERMINANTAL ASSIGNMENT PROBLEM AND APPROXIMATE DECOMPOSABILITY

The exterior equation an n—dimensional vector space over F, is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of ) is characterised by the Quadratic Pliicker Relations (QPR). An alternative new test for decomposability of is given, in terms of the rank properties of the Grassmann matrix, , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if dim = m, where . If is decomposable, then the solution space is simply defined by . The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties are linked to the singular values of .