For arbitrary $n\times n$ matrices M, I am trying to solve the integral
$\int_{\|v\| = 1} v^T M v.$
Solving this integral in a few low dimensions (by passing to spherical coordinates) suggests the answer in general to be $\frac{A\,\mathrm{tr}(M)}{n}$ where $A$ is the surface area of the $(n-1)$-dimensional sphere. Is there a nice, coordinate-free approach to proving this formula?