I am trying to produce an example of a (necessarily non-normal) matrix that has only eigenvalues with positive real part, but whose numerical range contains elements with strictly negative real part. Of course, I could take the standard counterexample $$ B=\begin{pmatrix}0 & 1\\ 0 & 0\end{pmatrix} $$ and simply change its sign.

However, for some reason I need a matrix whose columns (but not rows!) sum up to 0 - indeed, I am looking for a Kirchhoff matrix of a graph, in the jargon of Tutte - and I am having some difficulty finding a matrix with this property. Is there any theorem that forbids such a behaviour? I am aware of results on the numerical range of doubly stochastic matrices, but nothing about simply stochastic matrices or matrices of the form I am considering.