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

_{m}A_{n} = _{m}U_{p} _{p}S_{p} _{p}V_{n}′

The diagonal matrix S contains the *p* singular values of A, ordered such that

*s*_{1} ≥ *s*_{2} ≥ … ≥ *s _{p}* ≥ 0

The matrices U and V contain the left and right singular vectors of A, and are orthonormal: that is,

U′U = V′V = I_{p}

The smaller of U and V will be orthogonal. So, if A has more rows than columns, *m*>*n*, *p*=*n* and VV′=I_{p}.