Geodesic
Semitransitivity Working Group at LAW'05
The electronic journal of linear algebra, Tome 15 (2006), pp. 225-238.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: A set of matrices ${\cal S}\subseteq\Bbb M_n(\Bbb F)$ is said to be semitransitive if for any two nonzero vectors $x,y\in\Bbb F^n$, there exists a matrix $A\in{\cal S}$ such that either $Ax=y$ or $Ay=x$. In this paper various properties of semitransitive linear subspaces of $\Bbb M_n(\Bbb F)$ are studied. In particular, it is shown that every semitransitive subspace of matrices has a cyclic vector. Moreover, if $|\Bbb F|\ge n$, 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.
Classification : 15A30
Keywords: semitransitive subspaces of matrices, triangularizability, reducibility, minimality 
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     title = {Semitransitivity {Working} {Group} at {LAW'05}},
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Semitransitivity Working Group at LAW'05. The electronic journal of linear algebra, Tome 15 (2006), pp. 225-238. http://geodesic.mathdoc.fr/item/ELA_2006__15__a10/