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In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any m × n {displaystyle m imes n} matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics. Formally, the singular value decomposition of an m × n {displaystyle m imes n} real or complex matrix M {displaystyle mathbf {M} } is a factorization of the form U Σ V ∗ {displaystyle mathbf {USigma V^{*}} } , where U {displaystyle mathbf {U} } is an m × m {displaystyle m imes m} real or complex unitary matrix, Σ {displaystyle mathbf {Sigma } } is an m × n {displaystyle m imes n} rectangular diagonal matrix with non-negative real numbers on the diagonal, and V {displaystyle mathbf {V} } is an n × n {displaystyle n imes n} real or complex unitary matrix. The diagonal entries σ i {displaystyle sigma _{i}} of Σ {displaystyle mathbf {Sigma } } are known as the singular values of M {displaystyle mathbf {M} } . The columns of U {displaystyle mathbf {U} } and the columns of V {displaystyle mathbf {V} } are called the left-singular vectors and right-singular vectors of M {displaystyle mathbf {M} } , respectively.

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