  We present the analytical singular value decomposition of the stoichiometry
matrix for a spatially discrete reaction-diffusion system on a one dimensional
domain. The domain has two subregions which share a single common boundary.
Each of the subregions is further partitioned into a finite number of
compartments. Chemical reactions can occur within a compartment, whereas
diffusion is represented as movement between adjacent compartments. Inspired by
biology, we study both 1) the case where the reactions on each side of the
boundary are different and only certain species diffuse across the boundary as
well as 2) the case with spatially homogenous reactions and diffusion. We write
the stoichiometry matrix for these two classes of systems using a Kronecker
product formulation. For the first scenario, we apply linear perturbation
theory to derive an approximate singular value decomposition in the limit as
diffusion becomes much faster than reactions. For the second scenario, we
derive an exact analytical singular value decomposition for all relative
diffusion and reaction time scales. By writing the stoichiometry matrix using
Kronecker products, we show that the singular vectors and values can also be
written concisely using Kronecker products.
  Ultimately, we find that the singular value decomposition of the
reaction-diffusion stoichiometry matrix depends on the singular value
decompositions of smaller matrices. These smaller matrices represent modified
versions of the reaction-only stoichiometry matrices and the analytically known
diffusion-only stoichiometry matrix. Our results provide a mathematical
framework that can be used to study complex biochemical systems with metabolic
compartments. MATLAB code for calculating the SVD equations is available at
\url{www.github.com/MathBioCU/ReacDiffStoicSVD}.
