## Seminar 15/11/11: Spaces of matrices without non-zero eigenvalues in their field of definition

Spaces of matrices without non-zero eigenvalues in their field of definition

Rachel Quinlan

NUI Galway

Tuesday 15 November 2011

1pm, Room KA3-011, DIT Kevin Street

Abstract:

It was shown by M. Gerstenhaber in 1958 that for (almost) any field F, the maximum possible dimension of a linear subspace of M_{n}(F) in which every element is nilpotent is n(n-1)/2. In this talk we will use a duality relation involving the trace bilinear form to show that the same upper bound applies to the dimension in the more general situation of a subspace of M_{n}(F) in which no element has a non-zero eigenvalue that belongs to F. For example, the space of skew-symmetric n by n matrices over the field of real numbers possesses this latter property but does not consist of nilpotent matrices. The talk will include a discussion of the duality relation in a wider context involving affine spaces of square and rectangular matrices that have special rank properties and special covering properties. Only basic concepts in linear algebra and the theory of bilinear forms are involved.