Determinant of block-Toeplitz band matrices

Abstract

Some expressions are given for the determinant of an mn× mn block-Toeplitz band matrix L =[ L i−j ], with bandwidth ( p+ q+1) n< mn, in terms of the n× n generating matrix polynomial L(λ)=Σ p+ q j=0 λ j L p−j , det L -q ≠0. In the scalar case this yields formulas for the determinant expressed via the zeros of the generating (scalar) polynomial. The approach adopted in this work leans heavily on the recently developed spectral theory of matrix polynomials.