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. (C) 1998 Elsevier Science Inc. [References: 11]
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机译：我们研究了数值范围的形状。给出了作为椭圆盘的矩阵的数值范围的标准。应用该结果表明，不存在数值范围为椭圆（非圆）盘的3×3和4×4幂等矩阵。获得了n×n三对角矩阵具有椭圆数值范围的充分条件。检查尖点附近的数值范围的边界。最后，压缩了可约矩阵的数值范围，并讨论了其几何性质。 （C）1998 Elsevier Science Inc. [参考：11]
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