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

We explicitly determine the skew-symmetric eigenvectors and corresponding eigenvalues of the real symmetric Toeplitz matrices T = T(a, b, n) := (a + b vertical bar j - k vertical bar)(1 <= j,k <= n) of order n >= 3 where a, b is an element of R, b not equal 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 p(n)(delta; z) := (z(n+1) - delta z(n) - delta z + 1)/(z + 1) if n is even, and p(n)(delta; z) := z(n+1) - delta z(n) - delta z + 1 if n is odd, delta := 1 + 2/(2c + n - 1). (C) 2014 Elsevier Inc. All rights reserved.