The following question, arising from a construction involving
automorphisms of finite p-groups, was posed by F. Szechtman in a 2003
paper in the AMS Proceedings : for a vector space V of dimension n
over a field F, what is the minimum possible dimension of a
(non-linear) affine subspace of EndF(V)
that contains elements
annihilating all hyperplanes of V? This question is
equivalent, under a duality arising from the trace bilinear form on
Mn(F),
to the problem of determining the maximum possible dimension of a linear
subspace of
Mn(F) in which no element possesses a non-zero eigenvalue that
belongs to the field F. This talk will explain this duality and show how it
can be used to solve both of the problems mentioned above, independently of
the field under consideration.
The duality relation will then be explored in a wider context
involving affine
spaces of square and rectangular matrices that have special rank
properties
and special covering properties. One application that will be
discussed is to
the problem of characterizing partial matrices whose completions have
ranks
satisfying a prescribed lower bound.
The only background needed for this talk is basic linear algebra and a
little bit about bilinear forms.