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Given two real symmetric matrices $M,S$ is there a known answer for the Gaussian integral $\int d^Nz\frac{z^TMz}{z^TSz}$ where the integration is over N-dimensional Gaussian variable $z\sim N(\vec{0},I)$?

This can also be written as $E_z[\frac{z^TMz}{z^TSz}]$, so it seems like a very simple expression, but I could not find any result on this.

Uri Cohen
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  • I'm having a hard time understanding that integral. Are you asking about $E[\frac{Z^TMZ}{Z^TSZ}]$ where $Z$ is the $N$ dimensional random variable? – muaddib Aug 12 '15 at 10:41
  • @muaddib: yes, thanks – Uri Cohen Aug 12 '15 at 10:48
  • If $S$ is positive semi-definite it is enough to solve this for $S=I$ using a change-of-variables $Az=u$ for $A^TA=S$; in this case $E_z[\frac{z^TMz}{z^TSz}]=\frac{1}{|\det A|}\int d^{n}u\frac{u^{T}Qu}{u^{T}u}$ for $Q=A^{-T}MA^{-1}$ and $u\sim N(0,S)$. This seems like an integral of $u^TQu$ over the N-dimensional unit sphere, but I don't know how to calculate it. – Uri Cohen Aug 12 '15 at 14:15

2 Answers2

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If I'm not mistaken, Magnus [1986] ("The exact moments of a ratio of quadratic forms in normal variables") has your answer. See section 5, let s=1, for your case.

http://www.google.com/url?sa=t&source=web&cd=2&ved=0CCEQFjABahUKEwie29a07sLHAhXCNz4KHRj8Dg0&url=http%3A%2F%2Fannales.ensae.fr%2Fanciens%2Fn04%2Fvol4-05.pdf&rct=j&q=integral%20ratio%20quadratic%20forms&ei=XqfbVZ60DsLv-AGY-Lto&usg=AFQjCNH2P-MVDDyDAS0FjuvOUx1H7mrwCw

Geoffrey Evans
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Per @GeoffreyEvans answer the general case of the s-th moment $E_z[(\frac{z^TMz}{z^TSz})^s]$ for $z\sim N(\mu,\Sigma)$ is discussed by [Magnus 1986]. A (complex) closed-form answer is available there, but for $s=1$, $\mu=\vec{0}$ and $\Sigma =I$ which the question targeted it simplifies to

$$\mathbb{E}\left[\frac{z^{T}Mz}{z^{T}Sz}\right]=\int_{0}^{\infty}dt\prod_{j}\frac{1}{\sqrt{1+2t\lambda_{j}}}\sum_{i}\frac{\left[U^TMM^TU\right]_{i}^{2}}{1+2t\lambda_{i}}$$

where $S=U\Lambda U^T$ for orthonormal matrix $U$ and diagonal matrix $\Lambda$ with diagonal elements $\{\lambda_i\}$.

Uri Cohen
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