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The notion of dichotomous matrices is introduced as a natural generalizationof essentially Hermitian matrices. A criterion for arrowhead matrices to bedichotomous is established, along with necessary and sufficient conditions forsuch matrices to be unitarily irreducible. The Gau--Wu number (i.e., themaximal number $k(A)$ of orthonormal vectors $x_j$ such that the scalarproducts $\langle Ax_j,x_j\rangle$ lie on the boundary of the numerical rangeof $A$) is computed for a class of arrowhead matrices $A$ of arbitrary size,including dichotomous ones. These results are then used to completely classifyall $4\times4$ matrices according to the values of their Gau--Wu numbers.

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