BooksecrCollected Volumepp. 525–529

A Cayley–Hamiltonian theorem for periodic finite band matrices

  • Barry Simon

    California Institute of Technology, Pasadena, USA
A Cayley–Hamiltonian theorem for periodic finite band matrices cover

A subscription is required to access this book chapter.

Abstract

Let KK be a doubly infinite, self-adjoint matrix which is finite band (i.e. Kjk=0K_{jk} = 0 if jk>m|j-k| > m) and periodic (KSn=SnKKS^n = S^nK for some nn where (Su)j=uj+1(Su)_j = u_{j+1}) and non-degenerate (i.e. Kjj+m0K_{j j+m} \ne 0 for all jj). Then, there is a polynomial, p(x,y)p(x,y), in two variables with p(K,Sn)=0p(K,S^n) = 0. This generalizes the tridiagonal case where p(x,y)=y2yΔ(x)+1p(x,y) = y^2 - y \Delta(x) + 1 where Δ\Delta is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.