Abstract
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.
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