Solution of the determinantal assignment problem using the Grassmann matrices

Karcanias, N. & Leventides, J. (2016). Solution of the determinantal assignment problem using the Grassmann matrices. International Journal of Control, 89(2), pp. 352-367. doi: 10.1080/00207179.2015.1077525

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The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation (Formula presented.) where (Formula presented.) is an n −dimensional vector space over (Formula presented.) 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 are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of (Formula presented.), and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector (Formula presented.) are given in terms of the rank properties of the Grassmann matrix, (Formula presented.) of the vector (Formula presented.), which is constructed by the coordinates of (Formula presented.). It is shown that the exterior equation is solvable ((Formula presented.) is decomposable), if and only if (Formula presented.) where (Formula presented.); the solution space for a decomposable (Formula presented.), is the space (Formula presented.). 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 are 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.

Item Type: Article
Additional Information: This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Control on 2 Sept 2015, available online:
Uncontrolled Keywords: Frequency Assignment, Control Theory, Linear and Multilinear Algebra
Subjects: Q Science > QA Mathematics
Divisions: School of Engineering & Mathematical Sciences > Engineering

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