I am reading the paper Hierarchical Clustering of a Mixture Model by Goldberger et al.
On the page 2 of this paper they define the following function:
$ d\big(\mathcal N(\mathbb\mu_1, \Sigma_1),\;\mathcal N(\mathbb\mu_2, \Sigma_2)\big) = \frac{1}{2}\Big( \log\frac{\det\Sigma_2}{\det\Sigma_1} + \mathrm{tr}\big( \Sigma^{-1}_2\Sigma_1 \big) +(\mathbb\mu_1 - \mathbb\mu_2)^\top \Sigma_2^{-1}(\mathbb\mu_1 - \mathbb\mu_2) \Big) $
as the distance between the two multivariate Normal distributions: $\;\mathcal N(\mathbb\mu_1, \Sigma_1),\;\mathcal N(\mathbb\mu_2, \Sigma_2)$.
Since they define it as a distance, it implies that it must satisfy symmetry. But I don't see how the function above is symmetric.
In particular, I don't see how
$ \log\frac{\det Q}{\det P} + \mathrm{tr}\big( Q^{-1} P \big) + (\mathbf x - \mathbf y)^\top Q^{-1}(\mathbf x - \mathbf y) = \log\frac{\det P}{\det Q} + \mathrm{tr}\big( P^{-1} Q \big) + (\mathbf y - \mathbf x)^\top P^{-1}(\mathbf y - \mathbf x) $
is true for all $\;x,y\, \in\, \mathbb R^d,$ and symmetric positive definite matrices $\;P,Q\, \in\, \mathbb R^{d\times d}$.