Efficient Matrix Decomposition for High-Dimensional Structured Systems: Theory and Applications

In this paper, we introduce a novel matrix decomposition method, referred to as the \( D \)-decomposition, designed to improve computational efficiency and stability for solving high-dimensional linear systems. The decomposition factorizes a matrix \( A \in \mathbb{R}^{n \times n} \) into three matrices \( A = PDQ \), where \( P \), \( D \), and \( Q \) are structured to exploit sparsity, low rank, and other matrix properties. We provide rigorous proofs for the existence, uniqueness, and stability of the decomposition under various conditions, including noise perturbations and rank constraints. The \( D \)-decomposition offers significant computational advantages, particularly for sparse or low-rank matrices, reducing the complexity from \( O(n^3) \) for traditional decompositions to \( O(n^2 k) \) or better, depending on the structure of the matrix. This method is particularly suited for large-scale applications in machine learning, signal processing, and data science. Numerical examples demonstrate the method's superior performance over traditional LU and QR decompositions, particularly in the context of dimensionality reduction and large-scale matrix factorization.