A complete description of 4-by-4 matrices $\left[\begin{array}{cc}\alpha I& C\\ D& \text{β}I\end{array}\right]$, with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two non-concentric ellipses is given. This result is obtained by reduction to the leading special case in which $C-D^{\ast }$ also is a scalar multiple of the identity. In particular cases when in addition $\alpha -\text{β}$ is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.

中文翻译：

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在某些双椭圆数值范围的4×4矩阵上

4 x 4矩阵的完整描述 $\left[\begin{array}{cc}\alpha \mathrm{一世}& C\\ d& \text{β}\mathrm{一世}\end{array}\right]$给出了标量2×2对角线块，其数值范围是两个非同心椭圆的凸包。通过将最主要的特殊情况简化为$C-d^{\ast }$也是标识的标量倍数。在特殊情况下$\alpha -\text{β}$是实数还是纯虚数，结果采用特别简单的形式。提供了对等矩阵的应用。