An interlacing theorem for tridiagonal matrices

Abstract

If A is an n × n matrix and if S ⊂{1,...,n}, then let A(S) denote the principal submatrix of A formed by rows and columns in S. If A, B are n × n matrices, then let η(A, B) = Σ_sdet A(S) det B(S^t) where the summation is over all subsets of {1,...,n}, where S' denotes the complement of S, and where, by convention det A(φ) = det B(φ) = 1. It has been conjectured that if A is positive definite and B hermitian, then the polynomial η(λA, - B) has only real roots. We prove this conjecture if n ≤ 3, and also for any n under the additional assumption that both A, B are tridiagonal. We derive some consequences, including a generalization of a majorization result of Schur for tridiagonal matrices.