Note: Sorry if my notation is slightly off, my background is engineering not pure maths and I'm used to (loose notation).
I have a multivariable normal distribution:
$$x \sim \mathcal{N}(\bar{x}, \Sigma_x)$$
where:
$$x=x_1,x_2,x_3,...,x_n$$ $$\bar{x}=\bar{x_1},\bar{x_2},\bar{x_3},...,\bar{x_n}$$ $$\Sigma_x=\begin{bmatrix} \sigma_{x_1}^2 & cov(x_1,x_2) & ... & cov(x_1,x_n) \\ cov(x_2,x_1) & \sigma_{x_2}^2 & ... & ... \\ ... & ... & ... & ... \\ cov(x_n,x_1) & ... & ... & \sigma_{x_n}^2 \end{bmatrix}$$
If i were to combine 2 of these variables, $$ x_n, x_{n-1}$$ Such that: $$\bar{x_{new}} = w_n*\bar{x_n}+w_{n-1}*\bar{x_{n-1}}$$ $$\sigma_{x_{new}}^2 = w_n^2\sigma_{x_{n}}^2 + w_{n-1}^2\sigma_{x_{n-1}}^2 + 2w_nw_{n-1}cov(n,n_{-1})$$
I have the mean and variance of my new combined variable, but how does the covariance of this new variable update with each other variable? I.E. what is each new $$cov(x_{new},x_i),cov(x_i,x_{new})$$ for $$i -> n-2 $$
I am willing to update any notation if someone believes this is wildly unclear
Hint :
Recall that for random variables $X,Y,Z$ and constants $c$ we have
$$ \text{Cov}(X + Y,Z) = \text{Cov}(X,Z) + \text{Cov}(Y,Z)$$ $$ \text{Cov}(cX,Y) =c \cdot \text{Cov}(X,Y).$$