ELA, Volume 4, pp. 32-38, August 1998, abstract.

Z-pencils

Judith J. McDonald, D. Dale Olesky, Hans Schneider,
Michael J. Tsatsomeros, and P. van den Driessche
 

The matrix pencil (A,B)={tB-A | t in C} is considered under the 
assumptions that A is entrywise nonnegative and B-A is a 
nonsingular M-matrix. As t varies in [0,1], the Z-matrices tB-A 
are partitioned into the sets L_s introduced by Fiedler and 
Markham. As no combinatorial structure of B is assumed here, 
this partition generalizes some of their work where B=I.  Based 
on the union of the directed graphs of A and B, the combinatorial
structure of nonnegative eigenvectors associated with the largest 
eigenvalue of (A,B) in [0,1) is considered.