Singular Value Decomposition Using the Jacobi Pass-Through Method Algorithm

In linear algebra, singular value decomposition (SVD) is a crucial matrix factorization technique that decomposes any matrix into three component matrices: the left singular vector matrix, the singular value matrix, and the right singular vector matrix. The singular value matrix is a diagonal matrix whose diagonal elements represent the matrix's singular values. The Jacobi pass-through method algorithm is an optimized approach for computing singular value decomposition that leverages the advantages of the standard Jacobi algorithm while significantly improving computational efficiency. This optimization makes singular value decomposition more practical and feasible for real-world applications by implementing a threshold-based mechanism that focuses processing only on matrix elements exceeding a certain magnitude, thereby reducing unnecessary computations. The algorithm typically involves iterative orthogonal transformations through Jacobi rotations, where the implementation would include functions for element threshold checking, rotation angle calculation, and matrix update operations.