Geometry of higher-rank numerical ranges
Geometry of higher-rank numerical ranges
复制标题
高阶数值范围的几何
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
D. Kribs
中科院分区：
文献类型：
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作者：
Man;Michael Giesinger;J. Holbrook;D. Kribs
我们考虑任意N × N矩阵的高秩数值域的几何方面。特别感兴趣的是凸性问题和Toeplitz-Hausdorff定理的可能扩展。我们推导出一些减少，并获得部分结果的一般问题。我们还进行图形和计算实验。补充证据：在接受这篇论文后，我们的主题迅速发展。首先，Hugo Woerdeman将命题2.4和定理2.12与代数Riccati方程理论相结合，建立了高阶数值域的凸性。参见Woerdeman，H.，2007年，高阶数值域是凸的，线性和多线性代数，出现。随后，Chi-Kwong Li和Nung-Sing Sze采用了一种不同的方法，不仅产生凸性，而且还提供了重要的额外见解。参见Li，C.- K.和Sze，N.-美国，2007年，Canonical forms，higher rank numerical ranges，totally isotropic subspaces，and matrix equations，预印本。另见Li，C.- K.，潘耀国T.，和Sze，N.-美国，2007，高阶数值范围非空的条件，预印本。
We consider geometric aspects of higher-rank numerical ranges for arbitrary N × N matrices. Of particular interest is the issue of convexity and a possible extension of the Toeplitz–Hausdorff Theorem. We derive a number of reductions and obtain partial results for the general problem. We also conduct graphical and computational experiments. Added in proof: Following acceptance of this paper, our subject has developed rapidly. First, Hugo Woerdeman established convexity of the higher-rank numerical ranges by combining Proposition 2.4 and Theorem 2.12 with the theory of algebraic Riccati equations. See Woerdeman, H., 2007, The higher rank numerical range is convex, Linear and Multilinear Algebra, to appear. Subsequently Chi-Kwong Li and Nung-Sing Sze followed a different approach that not only yields convexity but also provides important additional insights. See Li, C.-K. and Sze, N.-S., 2007, Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations, preprint. See also Li, C.-K., Poon, Y.-T., and Sze, N.-S., 2007, Condition for the higher rank numerical range to be non-empty, preprint.