Abstract
AbstractWe introduce the notion of the slot length of a family of matrices over an arbitrary field
${\mathbb {F}}$
. Using this definition it is shown that, if
$n\ge 5$
and A and B are
$n\times n$
complex matrices with A unicellular and the pair
$\{A,B\}$
irreducible, the slot length s of
$\{A,B\}$
satisfies
$2\le s\le n-1$
, where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of
$n\times n$
matrices, where
$n\ge 5$
, is at most
$n-1$
. The slot length of a family of rank-one complex matrices can be equal to n.
Publisher
Cambridge University Press (CUP)