#StackBounty: #normal-distribution #circular-statistics Expected ratio of x'Ax and x'AAx on a unit sphere?

Bounty: 50

Suppose $A$ is symmetric positive definite matrix. Is there a nice expression for the first moment of the following quantity?

$$frac{x^TAx}{x^TA^2 x}$$

Where $x$ is distributed as $text{Normal}(0,I_n)$. This is the ratio of two quadratic forms evaluated on the surface of the sphere.

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When $A$ has eigenvalues $langle 1, frac{1}{2}rangle$, this expectation is equal to $frac{4}{3}$

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