Toeplitz antisymmetric tridiagonal matrix

For $M\ge 1$, the shifted skew-symmetric matrix $aI+bK$, where $K$ has zero diagonal, superdiagonal $1$, and subdiagonal $-1$, is a tridiagonal Toeplitz matrix. It is itself skew-symmetric only when $a=0$ over the real numbers or complex numbers. Its eigenvalues areAll such matrices share an orthonormal eigenbasis after the same diagonal unitary change from the discrete sine basis.