ELA, Volume 7, pp. 182-190, December 2000, abstract

Separable characteristic polynomials of pencils and property L

John Maroulas, Panayiotis J. Psarrakos, and Michael J. Tsatsomeros

The condition (SC): det(I-sA-tB) = det(I-sA) det(I-tB) for all 
scalars s,t, has naturally and long been connected to eigenvalue 
properties of the matrix pair A,B. In particular, Taussky used the 
notion of property L to generalize the Craig-Sakamoto Theorem by 
showing that when A and B are normal, (SC) is equivalent to AB=0. 
The relation of (SC) to the eigenspaces of A, B and sA+tB is examined 
in order to obtain necessary and/or sufficient conditions in terms 
of eigenspaces and space decompositions. A general criterion for (SC) 
based on the spectrum of the n by n matrix polynomial 
lambda^{2n+1} I - lambda^{2n} A - B  is also presented.