= Toeplitz antisymmetric tridiagonal matrix
{c}
For $M\ge1$, 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> are
$$
\lambda_j=a+2ib\cos\frac{j\pi}{M+1},
\qquad 1\leq j\leq M.
$$
All such <matrices> share an orthonormal <eigenbasis> after the same diagonal unitary change from the discrete sine basis.
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