= Solution
A spatial <finite difference method> and time discretization for a linear evolution problem give a recurrence $U^{n+1}=B_{h,k}U^n$. To obtain <numerical stability> on $0\le nk\le T$, one needs a bound $\|B_{h,k}^n\|\le C_T$ uniform in the permitted grid sizes and time steps. More generally a growth estimate $\|B_{h,k}^n\|\le C e^{c nk}$ is sufficient. The <finite-time stability versus power boundedness> distinction matters: growth per step of $1+O(k)$ can be compatible with the latter finite-time estimate, whereas unrestricted geometric growth at a fixed <Courant number> is not.
The eigenvalue method diagonalizes $B_{h,k}$ into modes. If $B_{h,k}$ is a <normal matrix>, the <spectral theorem> gives $\|B_{h,k}^n\|_2=\max_j|\mu_j|^n$. Thus $|\mu_j|\le1$ gives contraction, and $|\mu_j|\le e^{ck}$ gives the preceding growth estimate. A <diagonalizable matrix> $B=V\Lambda V^{-1}$ has $\|B^n\|_2\le\|V\|_2\|V^{-1}\|_2\max_j|\mu_j|^n$, so its <condition number> must remain bounded uniformly as the mesh changes. A stable-looking list of <eigenvalues> alone is insufficient.
Indeed,
$$
B=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad B^n=\begin{pmatrix}1&n\\0&1\end{pmatrix}.
$$
Both <eigenvalues> are one, but the <Jordan block> gives growth proportional to $n\sim T/k$, violating uniform finite-time <numerical stability>. A <nonnormal Fourier amplification matrix> can similarly exhibit large amplification even with its <eigenvalues> inside the <unit disk>. <Eigenvector> conditioning and the full powers of the amplification operator are essential.
For constant-coefficient schemes on a periodic or infinite grid, a <Fourier transform> diagonalizes scalar translation-invariant stencils. This yields <von Neumann stability analysis> using the <Fourier amplification symbol> $G(\theta)$. The <discrete Parseval identity> turns a bound on $G(\theta)^n$ into a discrete <L2 norm> estimate. For systems, the symbol is a <matrix>; a <uniform power bound for matrix Fourier symbols> is required, rather than a bound on its <spectral radius> alone.
For the <heat equation>, a <central finite difference> in space and the <explicit Euler method> in time give
$$
G(\theta)=1-4r\sin^2(\theta/2),\qquad r=k/h^2.
$$
The scalar condition $|G|\le1$ for every frequency is exactly \b[$0\le r\le1/2$]. With zero <Dirichlet boundary conditions>, the spatial <matrix> is <symmetric> and diagonalized by sine modes, leading to the same uniform sufficient restriction. By contrast, the <Backward Euler diffusion scheme> has $G(\theta)=[1+4r\sin^2(\theta/2)]^{-1}$ and is stable for every $r\ge0$.
For the <advection equation> $u_t+a u_x=0$, $a>0$, centered space with the <explicit Euler method> gives $G=1-i\nu\sin\theta$, $\nu=ak/h$, and $|G|^2=1+\nu^2\sin^2\theta$. At fixed positive <Courant number>, repeated powers are unbounded as the mesh is refined. The <upwind finite difference scheme> instead has
$$
G=1-\nu+\nu e^{-i\theta},\qquad |G|^2=1-2\nu(1-\nu)(1-\cos\theta),
$$
which gives contraction precisely for \b[$0\le\nu\le1$]. This comparison shows why the spatial stencil and time method must be analyzed together.
Boundary treatment can destroy the normality or mode decomposition available on a periodic grid, so a finite-interval scheme needs its own uniform operator estimate. Multilevel time schemes likewise require the <root condition for a multistep method> and uniform bounds for the associated amplification matrices; simple scalar root tests do not establish these bounds for arbitrary systems. Once <consistency of a numerical method>, a well-posed linear evolution problem and the required <numerical stability> estimate hold, the <Lax equivalence theorem> gives <numerical convergence>. Eigenvalues provide a powerful route to that estimate when the accompanying eigenvector and boundary analysis is justified.
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