The matrix cookbook contains formulas for the product of two multivariate Gaussians, but doesn't appear to contain formulas for the quotient of two Gaussians.
$$ \frac{\mathcal{N}(\mathbf{m}_1, \Sigma_1)}{\mathcal{N}(\mathbf{m}_2, \Sigma_2)} = ~?? $$
Note that I'm not looking for the quotient of two random variables (answered here), I just care about the quotient of two PDFs.
Context: I'm trying to derive factor analysis from Roweis and Ghahramani (1999). They apply Bayes rule:
$$ p(\mathbf{x} \mid \mathbf{y}) = \frac{p(\mathbf{y} \mid \mathbf{x}) p(\mathbf{x})}{p(\mathbf{y})} \\ =\frac{\mathcal{N}(C\mathbf{x}, R)\mathcal{N}(0, I)}{\mathcal{N}(0,CC^T + R)} \\ = \frac{\mathcal{N}((R^{-1} + I)^{-1} (R^{-1} C \mathbf{x}), (R^{-1} + I)^{-1})}{\mathcal{N}(0,CC^T + R)} \\ \vdots \\ ??? \\ \vdots \\ = \mathcal{N}(\boldsymbol{\beta} \mathbf{y}, I - \boldsymbol{\beta} C), \quad \boldsymbol{\beta} = C^T (C C^T + R)^{-1} $$
I used the matrix cookbook for the third equality. R is a covariance matrix without any special assumptions at this point in the paper.