The trace as an average over the unit sphere of a normed space with a 1-symmetric basis

We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis. Our result is new even in the case of ${\mathrm{\ell }}_p$-norms in ${\mathbb{R}}^N$ for $p\mathrm{\ne }2$.