= Solution
After spatial discretization, a linear time-dependent PDE produces an update
$$
u^{n+1}=Q_hu^n.
$$
Stability over $0\leq n\Delta t\leq T$ requires a bound $\lVert Q_h^n\rVert\leq C_T$ independent of the mesh. If $Q_h$ is normal, the <spectral theorem for normal operators> gives
$$
\lVert Q_h^n\rVert_2=\max_j|\lambda_j(Q_h)|^n,
$$
so eigenvalue analysis is decisive. More generally, if $Q_h=X_h\Lambda_hX_h^{-1}$, then
$$
\lVert Q_h^n\rVert\leq
\kappa(X_h)\max_j|\lambda_j|^n.
$$
Thus eigenvalues suffice only when the eigenvector condition numbers are uniformly bounded and unit-circle eigenvalues are semisimple. Defective or increasingly nonnormal matrices can have large powers even though every eigenvalue lies in the unit disk.
For a constant-coefficient scheme on the whole line or a periodic grid, the <discrete Fourier transform> diagonalizes translation-invariant difference operators. This is <Von Neumann stability analysis>: insert $u_m^n=G(\theta)^ne^{im\theta}$ and require $|G(\theta)|\leq1$. Its advantages are simplicity, sharp mesh restrictions, and direct identification of unstable wavelengths. Its limitations are boundaries, variable coefficients, nonlinearities, and nonnormality; frozen-coefficient Fourier analysis then gives at most local evidence.
As a successful implicit example, the <Backward Euler diffusion scheme> has
$$
Q_h=(I-\Delta tD_h)^{-1},
$$
where the periodic or homogeneous-Dirichlet discrete Laplacian $D_h$ is symmetric negative semidefinite. Its eigenvalues are $(1-\Delta t\lambda_j(D_h))^{-1}\in(0,1]$, proving unconditional discrete-$2$-norm stability.
For a failure, consider explicit upwinding for $u_t+a u_x=0$ on a finite inflow grid:
$$
u_j^{n+1}=(1-\mu)u_j^n+\mu u_{j-1}^n,
\qquad u_0^n=0.
$$
Its lower-triangular <nonnormal upwind amplification matrix> has only the eigenvalue $1-\mu$, so eigenvalues alone incorrectly suggest stability for $0\leq\mu\leq2$. For $1<\mu<2$, however, interior alternating data are amplified by $|1-2\mu|>1$ before the boundary is felt, and the matrix powers have no mesh-uniform bound. The correct range is $0\leq\mu\leq1$. This example isolates the missing hypothesis: spectral radius does not control powers of a nonnormal matrix family.
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