ELA, Volume 11, pp. 103-114, April 2004, abstract.

Eigenvalue Multiplicities of Products of Companion Matrices

Eric S. Key and Hans Volkmer

It is shown that under suitable conditions an eigenvalue of a 
product of companion matrices has geometric multiplicity equal 
to one. The result is used to show that for a class of Random 
Walks in Periodic Environments recurrence is equivalent to a 
product of companion matrices having 1 as an eigenvalue of
algebraic multiplicity greater than one.