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Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol \Sigma_1|$, where $|\cdot|$ is the determinant. Is there any relationship between $\boldsymbol x^\prime \boldsymbol \Sigma_2 \boldsymbol x$ and $\boldsymbol x^\prime \boldsymbol \Sigma_1 \boldsymbol x$?

Or $\boldsymbol x^\prime \boldsymbol \Sigma_2^{-1} \boldsymbol x$ and $\boldsymbol x^\prime \boldsymbol \Sigma_1^{-1} \boldsymbol x$? Or at least, $\text{E} (\boldsymbol x^\prime \boldsymbol \Sigma_2^{-1} \boldsymbol x)$ and $\text{E}(\boldsymbol x^\prime \boldsymbol \Sigma_1^{-1} \boldsymbol x)$?

Thank you.

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$|\boldsymbol \Sigma_2| < |\boldsymbol \Sigma_1|$, does not imply anything about the ordering of the quadratic forms $\boldsymbol x^\prime \boldsymbol \Sigma_2 \boldsymbol x$ and $\boldsymbol x^\prime \boldsymbol \Sigma_1 \boldsymbol x$, in general. (This ordering is sometimes called the Loewner ordering, written as $\boldsymbol \Sigma_2 \prec \boldsymbol \Sigma_1$ and it is implied if and only if $\boldsymbol \Sigma_2 - \boldsymbol \Sigma_1$ is positive-definite).

To see this, use the fact the determinant is the product of the eigenvalues, and consider diagonal matrices $$\Sigma_1 = \begin{bmatrix} 11 & 0 \\ 0 & 1 \end{bmatrix}$$ and $$\Sigma_2 = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$$ then for $x=\left[0, 1 \right]$ provides you with your counter-example. This example also demonstrates that $\boldsymbol \Sigma_1^{-1}$ and $\boldsymbol \Sigma_2^{-1}$ are not ordered by taking $x=\left[1, 0\right]$.

As far as expectations of the quadratic forms, you can use the higher-dimensional analog to the law of total variance to get that. From wiki, $$E(x^T \Lambda x) = \mbox{tr} (\Lambda \Sigma_1) + \mu^T \Lambda \mu.$$ So you would have that $E(\boldsymbol x^\prime \boldsymbol \Sigma_1^{-1} \boldsymbol x) = p$, and $E(\boldsymbol x^\prime \boldsymbol \Sigma_2^{-1} \boldsymbol x) = \mbox{tr}(\Sigma_1^{-1}\Sigma_2)$. Turning again to our counter example, this would be equal to $2 \frac{5}{11}>2$, so we see again that the answer is no.

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