Complex symmetric matrices with strongly stable iterates

We study complex-valued symmetric matrices. A simple expression for the spectral norm of such matrices is obtained, by utilizing a unitarily congruent invariant form. Consequently, we provide a sharp criterion for identifying those symmetric matrices whose spectral norm does not exceed one: such strongly stable matrices are usually sought in connection with convergent difference approximations to partial differential equations. As an example, we apply the derived criterion to conclude the strong stability of a Lax-Wendroff scheme.