线性代数 Resources

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Application Background Singular Value Decomposition (SVD) is an important matrix factorization method in linear algebra, extending the unitary diagonalization of normal matrices in matrix analysis. It has significant applications in signal processing, statistics, and other fields. SVD shares some similarities with eigenvector-based diagonalization of symmetric or Hermitian matrices, but despite their correlation, these two matrix decompositions have distinct differences. Key Technology A non-negative real number σ is a singular value of matrix M if there exist unit vectors u in Km and v in Kn such that: M = uσv^T where vectors u and v are respectively

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Iterative methods represent another class of techniques for solving linear algebraic equation systems, particularly effective for large sparse linear systems. These methods operate by designing specific iterative schemes that generate sequences of approximate solutions, which converge toward the exact solution when properly formulated. The implementation typically involves matrix-vector multiplications and residual calculations, requiring only O(n) storage for sparse systems. Key advantages include constant coefficient matrices throughout iterations, algorithmic simplicity, straightforward programming implementation, and reduced memory requirements compared to direct methods.

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