If you have a multivariate normal (MVN) distribution of format
$$ \begin{pmatrix} a \\ b \\ c \end{pmatrix} \sim MVN \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \sigma_{a,b} & \sigma_{a,c} \\ \sigma_{a,b} & \sigma^2_{b} & \sigma_{b,c} \\ \sigma_{a,c} & \sigma_{b,c} & \sigma^2_{c} \end{pmatrix} \right) $$
And you wish to find the multivariate normal distribution formed by taking functions of the elements of the original distribution - (in my case, I want to find distributions similar to):
$$ \begin{pmatrix} a \\ c-b \end{pmatrix} \sim MVN \left( \begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \sigma_{a,c-b} \\ \sigma_{a,c-b} & \sigma^2_{c-b} \end{pmatrix} \right) $$
Some elements appear straightforward:
$$ \begin{aligned} \sigma_a &= \sigma_a \\ \sigma_{c-b}&= \sqrt{\sigma^2_{b}+\sigma^2_{c}-2\sigma_{b,c}} \end{aligned} $$
However, could anyone give an example / point me to a paper or textbook that explains how to find the covariance between an element of a multivariate normal distribution, and a function of elements in the same multivariate normal distribution? So how you go about calculating the covariance between $a$ and $c-b$, which I'm denoting $\sigma_{a,c-b}$. I can find many examples for uncorrelated distributions, but I'm struggling to find a clear example where you have functions of correlated elements. Any pointers / advice would be appreciated.
EDIT - Answered:
Based on the link provided by d.k.o., the solution is to use affine transformations:
$$ X= \begin{pmatrix} a \\ b \\ c \end{pmatrix} \sim MVN \left( \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} \sigma^2_a & \sigma_{a,b} & \sigma_{a,c} \\ \sigma_{a,b} & \sigma^2_{b} & \sigma_{b,c} \\ \sigma_{a,c} & \sigma_{b,c} & \sigma^2_{c} \end{pmatrix} \right) $$
Let $Y=c+BX$. Set:
$$\begin{aligned} c&=\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\\ B&=\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 1 \\ 0 & 0 & 0 \end{pmatrix}\\ B^T&=\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{pmatrix} \end{aligned} $$ Then $Y$ has distribution $Y\sim MVN \left( c+B\mu , B\Sigma B^T \right)$ with $B\Sigma B^T$ equal to: $$ \begin{pmatrix} \sigma^2_a & \sigma_{a,c}-\sigma_{a,b} \\ \sigma_{a,c}-\sigma_{a,b} & \sigma_c^2+\sigma_b^2-2\sigma_{b,c} \end{pmatrix} $$