On geometric properties of the numerical range Original Research Article

We investigate the shape of the numerical range. A criterion for the numerical range of a matrix to be an elliptical disk is given. The result is applied to show that there exist neither 3-by-3 nor 4-by-4 nilpotent matrices whose numerical range is an elliptical (noncircular) disk. Sufficient conditions for n-by-n tridiagonal matrices to have elliptical numerical range are obtained. The boundary of the numerical range near a sharp point is examined. Finally, the numerical range of a reducible matrix is compressed, and its geometric properties are discussed.