Error propagation with covariance matrix I have two measurements $X = x \pm \sigma_{x}$ , $Y= y \pm \sigma_{y}$, where $x,y$ are the mean values of $X$ and $Y$ and $\sigma_{x},\sigma_{y}$ are their corresponding uncertainties. The covariance matrix ($\Sigma$) for them is the following:
\begin{equation}
\Sigma = \begin{pmatrix}
\sigma_{x}^{2} & \sigma_{xy}\\ 
\sigma_{yx} & \sigma_{y}^{2}
\end{pmatrix}
\end{equation}
Also I have a number $\textbf{Z} = z \pm \sigma_{z}$.
Suppose that I create a vector $V = \left \{ X,Y \right \}$ and I want to multiply the vector by the constant $\textbf{Z}$, what do I have to do in order to obtain the correct covariance matrix for V?
I know that first I have to do this multiplication $z\Sigma$, but I don't know how to do the error propagation. What should I do with  $\sigma_{z}$? Should I add in quadrature $\sigma_{z}$ to $\Sigma$?
Could you provide any reference to look for information about this topic?

You just need to apply the definition of Covariance to the product of the random variables:
Let $W = ZV$ be the random variable you are interested in, and note that by the independence of $Z$ and $V$ you have $w \equiv E[W]=E[Z]E[V] \equiv zv$ where $v=(x,y)$.
Now calculate directly the covariance of $W$:
$\Sigma_w \equiv Cov(W,W) = E[(W - E[W])(W - E[W])^T] = E[WW^T] - ww^T$
noting that (again using the independence of $Z$ and $V$)
$E[WW^T] = E[Z^2VV^T]=E[Z^2]E[VV^T]=(z^2+\sigma_z^2)(\Sigma + vv^T)$
you get
$\Sigma_w = (z^2+\sigma_z^2)\Sigma + \sigma_z^2vv^T$
$ \;\;\;\;\;\; =  (z^2+\sigma_z^2)\begin{pmatrix}
\sigma_{x}^{2} & \sigma_{xy}\\ 
\sigma_{xy} & \sigma_{y}^{2}
\end{pmatrix} + \sigma_z^2 \begin{pmatrix}
x^2 & xy\\ 
xy & y^2
\end{pmatrix}.$
