Abstract: To provide more efficient solutions for handling high-dimensional matrices and applying SVD(Singular Value Decomposition) in the context of big data, with the aim of accelerating data analysis and processing, how to quickly calculate the eigenvalues and eigenvectors ( singular value singular vectors) of high-dimensional matrices is studied. By studying random projection and Krylov subspace projection theory, six efficient calculation methods are summarized, making comparative analysis and related application research. Then, the six algorithms are applied, and the algorithms in related fields are improved. In the application of spectral clustering, the algorithm reduces the complexity of the core step SVD( Singular Value Decomposition), so that the optimized algorithm has similar accuracy to the original spectral clustering algorithm, but significantly shortens the running time. The calculation speed is more than 10 times faster than the original algorithm. When this work is applied in the field of image compression, it effectively improves the operation efficiency of the original algorithm. Under the condition of constant accuracy, the operation efficiency is improved by 1 ~ 5 times.

Key words: high-dimensional matrices, fast singular value decomposition ( SVD ), spectral clustering, image compression