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This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: http://openaccess.city.ac.uk/12469/ Link to published version: http://dx. Abstract The paper provides a direct solution to the Determinantal Assignment Problem (DAP) which unifies all frequency assignment problems of Linear Control Theory. The current approach is based on the solvability of the exterior equation 12 m …z v v v      where m i z v     , is an n  dimensional vector space over F which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of z , and it is in turn characterised by the set of Quadratic Plücker Relations (QPR) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector z are given in terms of the rank properties of the Grassmann matrix, () m n z  of the vector z, which is constructed by the coordinates of m z . It is shown that the exterior equation is solvable (z is decomposable), if and only if () m n dim z m  where () = { ()} mm n r n zz  ; the solution space for a decomposable z , is the space () { } m nz i z sp i m v    . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge-Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge-Grassmann matrix to the solution of exterior equations is examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP, (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist.

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