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 matrices . 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 (δ; z) := (z n+1 − δz n − δz + 1)/(z + 1) if n is even, and p n (δ; z) := z n+1 −δz n −δz +1 if n is odd, δ := 1+2/(2c+n−1)