Summary: The condition (SC): $det(I \Gamma sA\Gamma tB) = det(I \Gamma sA) det(I \Gamma tB)$ 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 suAEcient conditions in terms of eigenspaces and space decompositions. A general criterion for (SC) based on the spectrum of the n $\Theta n$ matrix polynomial * 2n+1 I $\Gamma * 2n$ A $\Gamma B$ is also presented.

15A15, 15A18, 15A22

characteristic polynomial, property L, craig-sakamoto theorem, matrix polynomial