The numerical range W(A) of an nxn matrix A is the totality of the scalar products <Ax,x> as x varies over all unit vectors in Cn The barycenter (center of mass) of the numerical range is defined geometrically as the center of mass of W(A) considered as a planar lamina with variable density and also as a limit of sample averages (<Ax1,x1>+...+<AxN,xN>)/N. Under a wide range the sampling schemes it is shown that the barycenter is the average of the spectrum (eigenvalues) of A weighted according to algebraic multiplicity which also equals tr(A)/n. The results of this paper justifies calling tr(A)/n the barycenter of W(A).
Broughton, Sean A.; Lautzenheiser, Roger G.; and Werne, Thomas, "The Barycenter of the Numerical Range of a Matrix" (2007). Mathematical Sciences Technical Reports (MSTR). 39.