Suppose we have a rectangular matrix A with m rows and n columns, and that p is the minimum of m and n. The singular value decomposition can be defined as
mAn = mUp pSp pVn′
The diagonal matrix S contains the p singular values of A, ordered such that
s1 ≥ s2 ≥ … ≥ sp ≥ 0
The matrices U and V contain the left and right singular vectors of A, and are orthonormal: that is,
U′U = V′V = Ip
The smaller of U and V will be orthogonal. So, if A has more rows than columns, m>n, p=n and VV′=Ip.