Location of Ritz values in the numerical range of normal matrices

Abstract

Let $\mu_1$ be a complex number in the numerical range $W(A)$ of a normal matrix $A$. In the case when no eigenvalues of $A$ lie in the interior of $W(A)$, we identify the smallest convex region containing all possible complex numbers $\mu_2$ for which $\begin{bmatrix}\mu_1& *\\0& \mu_2\end{bmatrix}$ is a $2$-by-$2$ compression of $A$.