[Abstract 6]

In this paper 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 exits neither 3-by-3
nor 4-by-4 nilponent matrix whose numerical range is an elliptical(noncircular) disk. Sufficient
conditions for n-by-n tridiagoanl 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.
Linear Algebra and Its Applications  Vol. 274, 1998, [ 389-410]