ELA, Volume 15, pp. 225-238, August 2006, abstract.
Semitransitive Subspaces of Matrices
Semitransitivity Working Group at LAW'05, Bled
A set of matrices S in M_n(F) is said to be semitransitive if for any two
nonzero vectors x,y in F^n, there exists a matrix A in S such that either
Ax=y or Ay=x. In this paper various properties of semitransitive linear
subspaces of M_n(F) are studied. In particular, it is shown that every
semitransitive subspace of matrices has a cyclic vector. Moreover, if
|F|>= n, it always contains an invertible matrix. It is proved that there
are minimal semitransitive matrix spaces without any nontrivial invariant
subspace. The structure of minimal semitransitive spaces and triangularizable
semitransitive spaces is also studied. Among other results it is shown that
every triangularizable semitransitive subspace contains a nonzero nilpotent.