Numerical range, maximal numerical range, normaloid matrices
The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in particular, when $W_0(A)=W(A)$. The set $W_0(A)$ is also described explicitly for matrices unitarily similar to direct sums of $2$-by-$2$ blocks, and some insight into the behavior of $W_0(A)$ is provided when $\mathcal L$ has codimension one.
Hamed, Ali N. and Spitkovsky, Ilya M..
"On the maximal numerical range of some matrices",
Electronic Journal of Linear Algebra,
Volume 34, pp. 288-303.
DOI: https://doi.org/10.13001/1081-3810, 1537-9582.3774