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.