# A Cayley–Hamiltonian theorem for periodic finite band matrices

### Barry Simon

California Institute of Technology, Pasadena, USA

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## Abstract

Let $K$ be a doubly infinite, self-adjoint matrix which is finite band (i.e. $K_{jk}=0$ if $∣j−k∣>m$) and periodic ($KS_{n}=S_{n}K$ for some $n$ where $(Su)_{j}=u_{j+1}$) and non-degenerate (i.e. $K_{jj+m}=0$ for all $j$). Then, there is a polynomial, $p(x,y)$, in two variables with $p(K,S_{n})=0$. This generalizes the tridiagonal case where $p(x,y)=y_{2}−yΔ(x)+1$ where $Δ$ is the discriminant. I hope Pavel Exner will enjoy this birthday bouquet.