In models with latent variables, the residual covariance matrix is calculated based on the matrix of latent variables regression coefficients formed by stacking the rows of \(\bm{\theta}_j\). That is, if we denote \(\bm{\Theta} = (\bm{\theta}_1 \ldots \bm{\theta}_p)'\), then the residual covariance and hence residual correlation matrix is calculated based on \(\bm{\Theta}\bm{\Theta}'\).
For multivariate abundance data, the inclusion of latent variables provides a parsimonious method of accounting for correlation between species. Specifically, the linear predictor,
$$\beta_{0j} + \bm{x}^\top_i\bm{\beta}_j + \bm{z}^\top_i\bm{\theta}_j$$
is normally distributed with a residual covariance matrix given by \(\bm{\Theta}\bm{\Theta}'\). A strong residual covariance/correlation matrix between two species can then be interpreted as evidence of species interaction (e.g., facilitation or competition), missing covariates, as well as any additional species correlation not accounted for by shared environmental responses (see also Pollock et al., 2014, for residual correlation matrices in the context of Joint Species Distribution Models).
The residual precision matrix (also known as partial correlation matrix, Ovaskainen et al., 2016) is defined as the inverse of the residual correlation matrix. The precision matrix is often used to identify direct or causal relationships between two species e.g., two species can have a zero precision but still be correlated, which can be interpreted as saying that two species do not directly interact, but they are still correlated through other species. In other words, they are conditionally independent given the other species. It is important that the precision matrix does not exhibit the exact same properties of the correlation e.g., the diagonal elements are not equal to 1. Nevertheless, relatively larger values of precision imply a stronger direct relationships between two species.
In addition to the residual correlation and precision matrices, the median or mean point estimator of trace of the residual covariance matrix is returned, \(\sum\limits_{j=1}^p [\bm{\Theta}\bm{\Theta}']_{jj}\). Often used in other areas of multivariate statistics, the trace may be interpreted as the amount of covariation explained by the latent variables. One situation where the trace may be useful is when comparing a pure LVM versus a model with latent variables and some predictors (correlated response models) -- the proportional difference in trace between these two models may be interpreted as the proportion of covariation between species explained by the predictors. Of course, the trace itself is random due to the MCMC sampling, and so it is not always guranteed to produce sensible answers!