Hurwitz-Radon matrices revisited: from effective solution of the Hurwitz matrix equations to Bott periodicity
Hurwitz-Radon matrices revisited: from effective solution of the Hurwitz matrix equations to Bott periodicity
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DOI：
10.1007/978-3-540-33791-1_12
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发表时间：
1994
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影响因子：
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通讯作者：
B. Eckmann
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文献类型：
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作者：
B. Eckmann
矩阵方程 Aj=—E, AjAk+ A^ Aj= 0, j, A:= 1,..., 5, j 7^ A: 由 Hurwitz 和 Radon 于 1920 年左右独立讨论*。多年来，解在数学和数学物理的许多部分中发挥了重要作用：作为椭圆微分算子的系数矩阵，在矢量场和纤维束的拓扑中，在组合分析中，在二次形式的复合问题中——这是最初的动机和起点，在该问题中，（复）矩阵必须是正交的。一个密切相关的变体是通过酉矩阵求解方程。在本次调查中，我们描述了一种有效构建解决方案的过程，首先在单一情况下，然后在正交情况下。重要的工具是“产品”，它将一切都简化为较小的 s 值；该产品同时与同伦和 iT 理论中著名的 Bott 周期性表现出非常密切的关系。我试图让非专业人士能够理解演示的大部分内容（除了那些涉及复杂和真实的 iC 理论的内容），以强调问题和解决方案的基本方面。我希望这在某种程度上符合彼得·希尔顿对各级数学教学做出的美好贡献的精神。
The matrix equations Aj=—E, AjAk+ A^ Aj= 0, j, A:= 1,..., 5, j 7^ A: were discussed independently by Hurwitz and Radon around 1920*. Solutions have played, during the years, an important role in many parts of mathematics and mathematical physics: as coefficient matrices of elliptic differential operators, in the topology of vector fields and fiber bundles, in combinatorial analysis, in the composition problem for quadratic forms—this was the original motivation and starting point, and in that problem the (complex) matrices have to be orthogonal. A closely related variant is to solve the equations by unitary matrices. In this survey we describe a procedure for constructing effectively the solutions, first in the unitary and from there in the orthogonal case. The essential tool is a" product" which reduces everything to small values of s; this product exhibits at the same time a very close relation with the well-known Bott periodicity in homotopy and iiT-theory.I have tried to keep large parts of the presentation (except for those concerning complex and real iC-theory) accessible to non-specialists, in order to emphasize the elementary aspects of the problem and the solution. I hope that this is somewhat in the spirit of Peter Hilton's beautiful contributions to the teaching of mathematics at various levels.