Crank-Nicolson centered-advection scheme on a finite interval

Let $D$ be the real skew-symmetric tridiagonal matrix with superdiagonal $1$ and subdiagonal $-1$. The centered-space Crank-Nicolson discretization of $u_t=u_x$ has amplification matrixThe eigenvalues of $D$ are $2i\cos (j\pi /(M+1))$, so those of $Q$ are Cayley transformsThey all have modulus one, and their common orthonormal eigenbasis makes $Q$ normal. Hence the method is stable for every $\mu >0$.