  Inspired by rational canonical forms, we introduce and analyze two
decompositions of dynamic programming (DP) problems for systems with linear
dynamics. Specifically, we consider both finite and infinite horizon DP
problems in which the dynamics are linear, the cost function depends only on
the state, and the state-space is finite dimensional but defined over an
arbitrary algebraic field. Starting from the natural decomposition of the
state-space into the direct sum of subspaces that are invariant under the
system's linear transformation, and assuming that the cost functions exhibit an
additive structure compatible with this decomposition, we extract from the
original DP problem two distinct families of smaller DP problems, each
associated with a system evolving on an invariant subspace of the original
state-space. We propose that each of these families constitutes a decomposition
of the original problem when the optimal policy and value function of the
original problem can be reconstructed from the optimal policies and value
functions of the individual subproblems in the family. We derive necessary and
sufficient conditions for these decompositions to exist both in the finite and
infinite horizon cases. We also propose a readily verifiable sufficient
condition under which the first decomposition exists, and we show that the
first notion of decomposition is generally stronger than the second.
