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|j − k|)1≤j,k≤n of order n ≥ 3 where a, b ∈ R, b 6= 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 detT , 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) := (z −δz−δz+1)/(z+1) if n is even, and pn(δ; z) := z −δz−δz+1 if n is odd, δ := 1+2/(2c+n−1).