Numerical ranges and matrix completions

There are two natural ways of defining the numerical range of a partial matrix. We show that for each partial matrix supported on a given pattern they give the same convex subset of the complex plane if and only if a graph associated with the pattern is chordal. This extends a previously known result (C.R. Johnson, M.E. Lundquist, Operator Theory: Adv. Appl. 50 (1991) 283–291) to patterns that are not necessarily reflexive and symmetric, and our proof overcomes an apparent gap in the proof given in the above-mentioned reference. We also define a stronger completion property that we show is equivalent to the pattern being an equivalence