Eigenvalue stability analysis of a finite difference method
= Eigenvalue stability analysis of a finite difference method
For a normal amplification matrix $Q$, the bound $\rho(Q)\leq1$ with semisimple unit-modulus eigenvalues controls every power $Q^n$. For a nonnormal family, eigenvalues alone do not control transient growth or give a mesh-uniform bound; eigenvector conditioning, pseudospectra, or a direct energy estimate is then needed.