,
The singular value decomposition (SVD) of a real
m x n matrix A is the factorization
,where U is an m x m orthogonal matrix, V is an n x n orthogonal matrix, and
is an m x n real diagonal matrix. The diagonal elements
of 
are ordered
so that
with r = min(m, n). If A is complex, its SVD is
,
where U is an m x m unitary matrix,
V is an n x n unitary matrix, and
and
is an m x n real diagonal matrix with ordered, nonnegative
diagonal elements.
For a proof of existence of the SVD, see [45].
The diagonal elements 
are
the singular values of A.
The first r columns of V are
right singular vectors of A, and
the first r columns of U are
its left singular vectors.
The number of nonzero singular values of A equals
the rank of A.