Linear maps preserving regional eigenvalue location

Abstract

Let M(n, C) be the vector space of n × n complex matrices and let G(r,s,t) be the set of all matrices in M(n, C) having r eigenvalues with positive real parts eigenvalues with negative real part and t eigenvalues with zero real part. In particularG(0,n,0) is the set of stable matrices. We investigate the set of linear operators on M(n, C) that map G(r,s,t) into itself. Such maps include, but are not always limited to similarities, transposition, and multiplication by a positive constant. The proof of our results depends on a characterization of nilpotent matrices in terms of matrices in a particular G(r,s,t), and an extension of a result about the existence of a matrix with prescribed eigenstructure and diagonal entries. Each of these results is of independent interest. Moreover, our char-acterization of nilpotent matrices is sufficiently general to allow us to determine the preservers of many other "inertia classes."

¹Partially supported by NSF Grant DMS-9000839 and ONR Grant N00014-90-J-1739.

²Partially supported by NSF Grant DMS-8900922.

³Partially supported by NSF Grant DMS-9000839.

⁴Partially supported by NSF Grant DMS-9101143.

⁵Partially supported by NSF Grant DMS-9007048. The fifth author gratefully acknowledges the assistance of the Department of Mathematics at the College of William and Mary, where this research was done while he was on sabbatical leave.

¹Partially supported by NSF Grant DMS-9000839 and ONR Grant N00014-90-J-1739.

²Partially supported by NSF Grant DMS-8900922.

³Partially supported by NSF Grant DMS-9000839.

⁴Partially supported by NSF Grant DMS-9101143.

⁵Partially supported by NSF Grant DMS-9007048. The fifth author gratefully acknowledges the assistance of the Department of Mathematics at the College of William and Mary, where this research was done while he was on sabbatical leave.

Notes

¹Partially supported by NSF Grant DMS-9000839 and ONR Grant N00014-90-J-1739.

²Partially supported by NSF Grant DMS-8900922.

³Partially supported by NSF Grant DMS-9000839.

⁴Partially supported by NSF Grant DMS-9101143.

⁵Partially supported by NSF Grant DMS-9007048. The fifth author gratefully acknowledges the assistance of the Department of Mathematics at the College of William and Mary, where this research was done while he was on sabbatical leave.