Fix an integer $m>2$, and denote by $\eS_m$ the space of real, symmetric $m\times m$ matrices. The group $\SO(m)$ of orthogonal transformations of $\bR^m$ acts on $\eS_m$ by conjugation. Fix a unit vector $\eta\in\bR^m$. We obtain a subgroup $\SO(m-1)$ of $\SO(m)$ consisting of transformations that fix the vector $\eta$. Denote by $\eQ_m$ the space of $\SO(m-1)$-invariant homogeneous quadratic polynomials on $\eS_m$. Then $\eQ_m$ is spanned by the quadratic polynomials

$$ A\mapsto \tr A^2,\;\; A\mapsto (\tr A)^2, \;\;A\mapsto |A\eta|^2, $$

$$ A\mapsto (A\eta,\eta)^2,\;\; A\mapsto (\tr A)(A\eta,\eta), $$

where $(-,-)$ denotes the canonical inner product on $\bR^m$.

There is a simple proof of this fact due to

**.**

*Robert Bryant*