On the real stability radius of a normal matrix

A well-known formula expresses the real stability radius of a real $n$-by-$n$ matrix $A$ as the minimax of a certain function of two parameters: a complex parameter $\lambda$, which varies along the boundary of the stability region, and a real parameter $\gamma$, which varies on the interval $(0,1]$. It is shown that, for a normal matrix $A$ with a known spectrum $\sigma(A)=\{\lambda_1,\dots,\lambda_n\}$, the maximization with respect to $\gamma$ can be replaced by a finite computation involving the eigenvalues $\{\lambda_1,\dots,\lambda_n\}$.