= Solution
Let $L=\operatorname{tridiag}(1,-2,1)$. The zero Dirichlet boundary values introduce no forcing terms, so the scheme is
$$
\boxed{Bu^{n+1}=Cu^n,\quad B=I-\frac\mu2L=\operatorname{tridiag}(-\mu/2,1+\mu,-\mu/2),\quad
C=I+\frac\mu2L=\operatorname{tridiag}(\mu/2,1-\mu,\mu/2).}
$$
The common orthogonal sine <eigenvectors> have $L$-eigenvalues $-4\sin^2[\pi j/(2(M+1))]$. Thus $B$ is invertible for every $\mu>0$, and each amplification <eigenvalue> is
$$
\boxed{g_j=\frac{1-2\mu\sin^2[\pi j/(2(M+1))]}{1+2\mu\sin^2[\pi j/(2(M+1))]},\qquad |g_j|\leq1.}
$$
The matrix $B^{-1}C$ is orthogonally diagonalizable, so its powers have <Euclidean norm> at most one, uniformly in the mesh and time step. Therefore \b[<Crank-Nicolson diffusion scheme> is stable for every $\mu>0$]. Negative amplification factors at large $\mu$ can create temporal sign oscillations, which do not contradict this <norm> stability.
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