Discriminant of Symmetric Matrices as a Sum of Squares and the Orthogonal Group
Discriminant of Symmetric Matrices as a Sum of Squares and the Orthogonal Group
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DOI：
10.1002/cpa.20353
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发表时间：
2011-04-01
影响因子：
3
通讯作者：
Domokos, Matyas
中科院分区：
文献类型：
--
作者：
Domokos, Matyas
证明了n × n真实的对称矩阵的判别式可以表示为平方和，其中被加数等于n次n元球谐函数空间的维数.应用正交群的表示理论，将3 × 3真实的对称矩阵的判别式表示为5个平方和，并证明了它不能写成小于5个平方和的判别式.证明了4 × 4真实的对称矩阵的判别式可写为7个平方之和。这些改进的结果库默从1843年和Borchardt从1846年。(C)2010 Wiley Periodicals，Inc.
It is proved that the discriminant of n x n real symmetric matrices can be written as a sum of squares, where the number of summands equals the dimension of the space of n-variable spherical harmonics of degree n. The representation theory of the orthogonal group is applied to express the discriminant of 3 x 3 real symmetric matrices as a sum of five squares and to show that it cannot be written as the sum of less than five squares. It is proved that the discriminant of 4 x 4 real symmetric matrices can be written as a sum of seven squares. These improve results of Kummer from 1843 and Borchardt from 1846. (C) 2010 Wiley Periodicals, Inc.