ON GEOMETRIC-PROPERTIES OF THE NUMERICAL RANGE

We investigate the shape of the numerical range. A criterion for the n umerical range of a matrix to be an elliptical disk is given. The resu lt is applied to show that there exist neither 3-by-3 nor 4-by-4 nilpo tent matrices whose numerical range is an elliptical (noncircular) dis k. Sufficient conditions for n-by-n tridiagonal matrices to have ellip tical numerical range are obtained. The boundary of the numerical rang e near a sharp point is examined. Finally, the numerical range of a re ducible matrix is compressed, and its geometric properties are discuss ed. (C) 1998 Elsevier Science Inc.