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Assume that the $p \times 1$ vector $\mathbf{y}$ has multivariate Normal distribution with $\mathbb{E}[\mathbf{y}] = \boldsymbol{\mu}$ and $\mathrm{V}[\mathbf{y}] = \boldsymbol{\Sigma}$. Let $\mathbf{A}$ and $\mathbf{G}$ be two known $p\times p$ constant matrices.

Define the quadratic form $Q = \mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y}$ and the vector $\mathbf{v} = \mathbf{G}\,\mathbf{y}$. Is there analytical formula for the covariance between $Q$ and $\mathbf{v}$?

That is how can I compute $\mathrm{Cov}(\mathbf{v}, Q)$ and $\mathrm{Cov}(Q, \mathbf{v})$?

I think the $\mathrm{Cov}(\mathbf{v}, Q) \neq \mathrm{Cov}(Q, \mathbf{v})$. Also, my attempt to find $\mathrm{Cov}(Q, \mathbf{v})$ is as follows:

\begin{align*} \begin{split} \mathrm{Cov}(\mathbf{v}, Q) &= \mathrm{Cov}(\mathbf{G}\,\mathbf{y}, \mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y}) \\ &= \mathbf{G}\,\mathrm{Cov}(\mathbf{y}, \mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y}) \\ &= \mathbf{G}\,\left\{\mathbb{E}\left[\mathbf{y}\,\mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y} \right] - \mathbb{E}\left[\mathbf{y}\right]\,\mathbb{E}[\mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y}] \right\} \end{split} \end{align*}

But I am not sure is there is a matrix expression for $\mathbb{E}\left[\mathbf{y}\,\mathbf{y}^\prime\,\mathbf{A}\,\mathbf{y} \right]$, because the argument is going to be a vector $p\times 1$, I cannot apply the $\mathrm{trace}$ function.

If anyone has any insights, suggestions, or solutions related to I would be grateful for your input.

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  • $\begingroup$ Do $A$ and $G$ have any special structure? i.e. are they invertible, positive semi-definite, etc.? $\endgroup$
    – mhdadk
    Aug 19, 2023 at 17:31
  • $\begingroup$ In my problem they are the first and second derivatives of a known function, so I can say that they are symmetric and invertible. $\endgroup$
    – andre
    Aug 19, 2023 at 17:42
  • $\begingroup$ Certainly there's a formula. Just write out any formula you like for covariance. For instance, the hard part is $$E[y^\prime Ay\, Gy]_n=\sum_{ijk} A_{ij}G_{nk}E[y_iy_jy_k].$$ $\endgroup$
    – whuber
    Aug 19, 2023 at 18:15
  • $\begingroup$ Please consider accepting my answer @andre $\endgroup$
    – dherrera
    Nov 30, 2023 at 21:22

1 Answer 1

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The answer to your question is given in the book "Quadratic Forms in Random Variables" from Mathai and Provost, page 74, Theorem 3.2d.2.

Let $y \in \mathbb{R}^p \sim \mathcal{N}(\mathbf{\mu}, \mathbf{\Sigma})$, $Q = \mathbf{y}'\mathbf{A}\mathbf{y}$ and $P = \mathbf{g}'\mathbf{y}$, with $\mathbf{A} \in \mathbb{R}^{p\times p}$ symmetric and $\mathbf{g} \in \mathbb{R}^p$.

Then, $Cov(Q,P) = 2 \mathbf{\mu}' \mathbf{A} \mathbf{\Sigma} \mathbf{g}$

In the formula above, $P \in \mathbb{R}$. This extends easily to your questions with $\mathbf{v} = \mathbf{G}\mathbf{y} \in \mathbb{R}^p$:

$Cov(Q,P) = \mathbf{\mu}' \mathbf{A} \mathbf{\Sigma} \mathbf{G}'$

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