The singular value decomposition (SVD) is a matrix factorization algorithm.

For m > n, the singular value decomposition of a real m × n matrix A is the factorization

The matrices in this factorization have the following properties:

U [m × n] and V [n × n]

are orthogonal matrices. The columns u_i of U =[u₁, ..., u_n] are the left singular vectors, and the columns v_i of V [v₁, ..., v_n] are the right singular vectors.

Σ [n × n] = diag (σ₁, ..., σ_n)

is a real, nonnegative, and diagonal matrix. Its diagonal contains the so called singular values σ_i, where σ₁ ≥ ... ≥ σ_n ≥ 0.

If m < n, the decomposition results in U [m × m] and V [n × m].

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