I have two sets of measurements, $x_i$ and $y_i$, both with the same number of elements, $N$. For each of these sets, I have known $N\times N$ covariance matrices, $\Sigma_x$ and $\Sigma_y$, representing the uncertainties in these measurements. The two sets of measurements have independent uncertainties with respect to each other. I want to take the element-wise quotients, $z_i = x_i / y_i$ and find the covariance matrix $\Sigma_z$. I know how to do this for single elements, but not sets of $N$ elements.
I know the standard propagation of error for a quotient, $\sigma^2_z/z^2 = \sigma_x^2 / x^2+ \sigma_y^2 / y^2$ (there's no correlation term because $x$ and $y$ are uncorrelated). I'm just not sure how this generalizes to the case of covariance matrices and vectors of measurements.
My intuition is that there must be a generalization that looks like $a \Sigma_z b = c \Sigma_x d + e \Sigma_y f$, where I need to find the forms of the matrices $a, b,c,d,e,$ and $f$, which are all $N \times N$ matrices somehow related to my measurement vectors, $x,y,$ and $z$.