Singular value decomposition

In linear algebra, the singular-value decomposition (SVD) is a factorization of a real or complex matrix. It is the generalization of the of a positive semidefinite normal matrix (for example, a symmetric matrix with positive eigenvalues) to any 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 formula_1 complex matrix is a factorization of the form formula_2, where is an formula_3 complex , formula_4 is an formula_1 with non-negative real numbers on the diagonal, and is an formula_6 complex unitary matrix. The diagonal entries formula_7 of formula_4 are known as the of . The columns of and the columns of are called the left-singular vectors and right-singular vectors of , respectively.
The singular-value decomposition can be computed using the following observations: