<P> 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 \ times n) matrix via an extension of the polar decomposition . It has many useful applications in signal processing and statistics . </P> <P> Formally, the singular - value decomposition of an m × n (\ displaystyle m \ times n) real or complex matrix M (\ displaystyle \ mathbf (M)) is a factorization of the form U Σ V ∗ (\ displaystyle \ mathbf (U \ Sigma V ^ (*))), where U (\ displaystyle \ mathbf (U)) is an m × m (\ displaystyle m \ times m) real or complex unitary matrix, Σ (\ displaystyle \ mathbf (\ Sigma)) is a m × n (\ displaystyle m \ times n) rectangular diagonal matrix with non-negative real numbers on the diagonal, and V (\ displaystyle \ mathbf (V)) is an n × n (\ displaystyle n \ times 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 . </P>

What is singular value decomposition of a matrix