Toeplitz matrix

A Toeplitz matrix is constant along each diagonal, so its entries have the form $T_{nm}=t_{n-m}$.

An $m\times m$ symmetric tridiagonal Toeplitz matrix has one constant diagonal $\alpha$ and equal constant subdiagonal and superdiagonal $\beta$. Its eigenvectors are the discrete sine transform vectors $q_j^{(k)}=\sin (jk\pi /(m+1))$, with eigenvalues $\alpha +2\beta \cos (k\pi /(m+1))$.

An $M\times M$ Toeplitz antisymmetric tridiagonal matrix with diagonal $a$, superdiagonal $b$, and subdiagonal $-b$ has eigenvaluesAll such matrices share an orthonormal eigenbasis after the same diagonal unitary change from the discrete sine basis.