Singular value decomposition (SVD)

The SVD factors a matrix into three: A = UDVᵀ. For A of shape m × n:

• U is m × m and orthogonal. Its columns are the left-singular vectors, the eigenvectors of AAᵀ.
• D is m × n and diagonal, though not necessarily square. Its diagonal entries are the singular values of A.
• V is n × n and orthogonal. Its columns are the right-singular vectors, the eigenvectors of AᵀA.

The nonzero singular values are the square roots of the eigenvalues of AᵀA (or, equally, of AAᵀ).

Unlike the eigendecomposition, which is not even defined for non-square matrices, every real matrix has an SVD. Its most useful feature: it partly generalizes matrix inversion to non-square matrices (see The Moore-Penrose pseudoinverse).