The Gau-Wu Number for $4\times 4$ and Select Arrowhead Matrices

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalar products $⟨A{x}_j,x_j⟩$ lie on the boundary of the numerical range of $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size, including dichotomous ones. These results are then used to completely classify all $4\times 4$ matrices according to the values of their Gau–Wu numbers.