对称矩阵为平方和与正交群的判别式
Discriminant of symmetric matrices as a sum of squares and the orthogonal group
摘要：
证明了$n\times n$实对称矩阵的判别式可以写成平方和，其中被和数等于$n$ -变次球谐函数$n$空间的维数.应用正交群的表示论，将3 × 3实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.本文证明了四乘四实对称矩阵的判别式可以写成七个平方之和。这些改善结果库默尔从1843年和Borchardt从1846年。
英文摘要：
It is proved that the discriminant of $n\times n$ 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 three by three real symmetric matrices as a sum of five squares, and to show that it can not be written as the sum of less than five squares. It is proved that the discriminant of four by four real symmetric matrices can be written as a sum of seven squares. These improve results of Kummer from 1843 and Borchardt from 1846.
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