Discriminant of symmetric matrices as a sum of squares and the orthogonal group

TL;DR

This paper proves that the discriminant of real symmetric matrices can be expressed as a sum of squares, with specific minimal numbers of squares for 3x3 and 4x4 matrices, improving historical results.

Contribution

It establishes the minimal number of squares needed to represent the discriminant of symmetric matrices and applies orthogonal group representation theory to these problems.

Findings

Discriminant of n×n symmetric matrices is a sum of squares with summands equal to the dimension of n-variable spherical harmonics.

For 3×3 matrices, the discriminant is a sum of five squares and cannot be fewer.

For 4×4 matrices, the discriminant is a sum of seven squares.

Abstract

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.