Pole Placement and Matrix Extension Problems : a Common

This paper studies a general inverse eigenvalue problem which generalizes many well-studied pole placement and matrix extension problems. It is shown that the problem corresponds geometrically to a so-called central projection from some projective variety. The degree of this variety represents the number of solutions the inverse problem has in the critical dimension. We are able to compute this degree in many instances, and we provide upper bounds in the general situation. POLE PLACEMENT AND MATRIX EXTENSION PROBLEMS: A COMMON POINT OF VIEW∗ MEEYOUNG KIM† , JOACHIM ROSENTHAL‡ , AND XIAOCHANG ALEX WANG§ SIAM J. CONTROL OPTIM. c © 2004 Society for Industrial and Applied Mathematics Vol. 42, No. 6, pp. 2078–2093 Dedicated to the memory of Meeyoung Kim, 1966–2001 Abstract. This paper studies a general inverse eigenvalue problem which generalizes many wellstudied pole placement and matrix extension problems. It is shown that the problem corresponds geometrically to a so-called central projection from some projective variety. The degree of this variety represents the number of solutions the inverse problem has in the critical dimension. We are able to compute this degree in many instances, and we provide upper bounds in the general situation. This paper studies a general inverse eigenvalue problem which generalizes many wellstudied pole placement and matrix extension problems. It is shown that the problem corresponds geometrically to a so-called central projection from some projective variety. The degree of this variety represents the number of solutions the inverse problem has in the critical dimension. We are able to compute this degree in many instances, and we provide upper bounds in the general situation.