Inverses, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entries

Abstract

We explicitly determine the skew-symmetric eigenvectors and corresponding eigenvalues of the real symmetric Toeplitz matricesT=T(a,b,n):=( a+b|j-k|)1≤j,k≤n of order n≥3 where a, b ∈ ℝ, b ≠0. The matrix T is singular if and only if c := a/b = -n-1/2. In this case we also explicitly determine the symmetric eigenvectors and corresponding eigenvalues of T. If T is regular, we explicitly compute the inverse T- 1, the determinant det T, and the symmetric eigenvectors and corresponding eigenvalues of T are described in terms of the roots of the real self-inversive polynomial pn(δ;z):=(zn+1- δzn-δz+1)/(z+1) if n is even, and pn(δ; z):=zn+1-δzn-δz+1 if n is odd, δ:=1+2/(2c+n-1). © 2014 Elsevier Inc.