Zero Assignment, Pole Placement and Matrix Extension Problems: A Common Point of View

The paper studies a general inverse eigenvalue problem which contains as special cases many well studied pole placement and matrix extension problems. It is shown that the studied problem corresponds on the geometric side to a central projection from some projective variety. The degree for this variety is computed in the critical dimension. 1 Introduction and motivational examples Let K be an arbitrary field and consider matrices of size E; A of size n \Theta n and matrices B; H of size n \Theta m. These matrices define the discrete dynamical Ex t+1 + Ax t +Hu t+1 +Bu t = 0: (1.1) Supported in part by NSF grant DMS-96-10389. Consider the vector space Matm\Thetan consisting of all m \Theta n matrices defined over K . Let L ae Matm\Thetan be a linear subspace of dimension d. This paper will be devoted to the following `constrained' pole placement question: Problem 1.1 Given an arbitrary monic polynomial '(s) 2 K [s] of degree n. Is there a feedback law of the form u t = F..