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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 modifie
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal neighbor and near
I show analytically that the average of $k$ bootstrapped correlation matrices rapidly becomes positive-definite as $k$ increases, which provides a simple approach to regularize singular Pearson correlation matrices. If $n$ is the number of objects an
The hierarchical SVD provides a quasi-best low rank approximation of high dimensional data in the hierarchical Tucker framework. Similar to the SVD for matrices, it provides a fundamental but expensive tool for tensor computations. In the present wor
In this paper we introduce the algorithm and the fixed point hardware to calculate the normalized singular value decomposition of a non-symmetric matrices using Givens fast (approximate) rotations. This algorithm only uses the basic combinational log
This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis.