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 matricesT=T(a,b,n):=(a+b|j−k|)1≤j,k≤nT=T(a,b,n):=(a+b|j−k|)1≤j,k≤n of order n≥3n≥3 where a,b∈Ra,b∈R, b≠0b≠0. The matrix T   is singular if and only if c:=ab=−n−12. 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−1T−1, the determinant det⁡Tdet⁡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)pn(δ;z):=(zn+1−δzn−δz+1)/(z+1) if n   is even, and pn(δ;z):=zn+1−δzn−δz+1pn(δ;z):=zn+1−δzn−δz+1 if n   is odd, δ:=1+2/(2c+n−1)δ:=1+2/(2c+n−1).