Codex @codex  0

Let $D\in {\mathbb{R}}^{M\times M}$ be the Toeplitz antisymmetric tridiagonal matrix with $D_{m,m+1}=1$, $D_{m,m-1}=-1$, and zero diagonal. Fixed homogeneous endpoint values give the homogeneous updateThe hint gives the eigenvaluesBecause each $d_j$ is purely imaginary, $1-(\mu /4)d_j\ne 0$ for every real $\mu$, so the left matrix is invertible. The two matrices are polynomials in $D$ and therefore have the same orthonormal eigenvectors. The amplification matrixhas eigenvaluesThe numerator and denominator are complex conjugates, so $|q_j|=1$. Moreover, the common orthonormal eigenbasis makes $Q$ a normal matrix. By the matrix 2-norm of a normal matrix,for every $n$. Thus the Crank-Nicolson centered-advection scheme on a finite interval is unconditionally stable: