Covariance matrix of element-wise quotient of two sets of measurements with known covariance matrices 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 $z$.
 A: Do the same you do to find the approximate solution
$$ \sigma^2_z/z^2 = \sigma_x^2 / x^2+ \sigma_y^2 / y^2 $$
which is base on approximating the function $f(x,y)= x/y $ by its first order Taylor series (around $x_0, y_0$ which we can take to be the expectations)
$$ f(x,y) \approx x_0/y_0 +f_x'(x_0,y_0) (x-x_0) + f_y'(x_0,y_0) (y-y_0) \\
  = x_0/y_0 + \frac1{y_0} (x-x_0) - \frac{x_0}{y_0^2} (y-y_0)
$$ and calculating the variance of this approximation gives the variance formula, after some manipulations. Now do the same but with covariance, and remember the bilinearity of covariance:
$$ \DeclareMathOperator{\C}{\mathbb{C}}
\C(z_i, z_j)\approx \C\left( \frac1{y_{i0}} (x_i-x_{i0}) - \frac{x_{i0}}{y_{i0}^2} (y_i-y_{i0}),
\frac1{y_{j0}} (x_j-x_{j0}) - \frac{x_{j0}}{y_{j0}^2} (y_j-y_{j0})  \right)
$$ After some manipulation that I leave for you this gives the approximation
$$\frac{\sigma_{zij}}{z_i z_j} = \frac{\sigma_{xij}}{x_i x_j} + \frac{\sigma_{yij}}{y_i y_j}
$$
If you want that can be written in matrix form. Let $D_x, D_y, D_z$ be diagonal matrices with the subscript vector along the diagonal.  Then
$$
 \Sigma_z= D_z \left\{ D_x^{-1}\Sigma_x D_x^{-1} + 
  D_y^{-1}\Sigma_y D_y^{-1} \right\} D_z
$$
(which is easy to see using the rules from Intuition for the Product of Vector and Matrices.)
