Integral around unit sphere of inner product 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?

As a function of $M$ your integral is linear, and is invariant under conjugation by orthogonal transformations ($C_R: M \mapsto R^{T} M R$).  Now the average of $C_R(M)$ over all orthogonal transformations $R$ (using Haar measure) is $\text{Tr}(M) I/n$ (it must be invariant under all $C_R$ so it is a multiple of $I$, and the trace is preserved). So the integral is the same as it would be for $\text{Tr}(M)I/n$, which is $\text{Tr}(M)/n$ times the area of the sphere. 
