On a conjecture regarding characteristic polynomial of a matrix pair

Abstract

For n-by-n Hermitian matrices A(> 0) and B, define

where the summation is over all subsets of {1,...,n}, S' is the complement of S, and by convention
det A(∅) = 1. Bapat proved for n = 3 that the zeros of η(λA, −B) and the zeros of η(λA(23), −B(23))
interlace. This result is generalized to a broader class of matrices.