Causal disentanglement seeks a representation of data involving latent variables that relate to one another via a causal model. A representation is identifiable if both the latent model and the transformation from latent to observed variables are unique.

In this paper, we study observed variables that are a linear transformation of a linear latent causal model. Data from interventions are necessary for identifiability: if one latent variable is missing an intervention, we show that there exist distinct models that cannot be distinguished.

Conversely, we show that a single intervention on each latent variable is sufficient for identifiability. Our proof uses a generalization of the RQ decomposition of a matrix that replaces the usual orthogonal and upper triangular conditions with analogues depending on a partial order on the rows of the matrix, with partial order determined by a latent causal model.

We corroborate our theoretical results with a method for causal disentanglement that accurately recovers a latent causal model.