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Singular value decomposition, often abbreviated as SVD, is a fundamental matrix factorization technique in linear algebra. It decomposes any given matrix into three other matrices. For a real matrix A of size m x n, its singular value decomposition is given by A = U * Sigma * V^T, where U is an m x m orthogonal matrix, Sigma is an m x n diagonal matrix containing non-negative real numbers on its diagonal, and V^T is the transpose of an n x n orthogonal matrix V. The diagonal entries of Sigma are called the singular values of A, and they are typically arranged in descending order. The columns of U are called the left singular vectors, and the columns of V are called the right singular vectors.

Singular value decomposition has wide-ranging applications across various fields. In image processing, it is used for image compression and noise reduction. In natural language processing, it forms the basis of techniques like Latent Semantic Analysis (LSA) for topic modeling and document analysis. It is also crucial in recommender systems for predicting user preferences and in principal component analysis (PCA) for dimensionality reduction and data visualization. The ability of SVD to capture the most significant information within a dataset makes it a powerful tool for data analysis and manipulation. The singular values provide a measure of the importance of each corresponding singular vector, allowing for the selection of the most relevant components of the data.