A not-so-normal mode decomposition

We provide a generalization of the normal mode decomposition for non-symmetric or locality constrained situations. This allows for instance to locally decouple a bipartitioned collection of arbitrarily correlated oscillators up to elementary pairs into which all correlations are condensed. Similarly, it enables us to decouple the interaction parts of multi-mode channels into single-mode and pair-interactions where the latter are shown to be a clear signature of squeezing between system and environment. In mathematical terms the result is a canonical matrix form with respect to real symplectic equivalence transformations.