Suppose $A \in R^{n \times n}$ is a symmetry matrix, I want to calculate the following thing: \begin{equation} \int_{\|x\|_2^2 = 1} x^TAx d\sigma(x) \end{equation}
i.e. I want to calculate the term $x^TAx$ over the unit sphere in $R^n$. What would I get?
Here $\sigma(x)$ is the standard Lebesgue measure on sphere.
My guess is something related to the eigenvalue of $A$, or maybe something related to $tr(A)$. But I don't know how to do the calculation.
Edit: Thanks for all the replies. Now I know how to deal with the case where I want to evaluate the mean of $x^TAx$. The trick here is to smartly diagonalize it.
However, I'm still confused about how to compute an integration on the sphere. The reason is that the thing I really want to evaluate is something like: $\int_{S^{n-1}}\frac{1}{x^TAx} d\sigma(x) $, where we can assume $A$ is positive definite. So in this case, we still need to face with the difficulty of tedious sphere coordinate transform. Is there any reference to this kind of integration? If we can not calculate its value, is there any method we can bound this thing?