A为m×n矩阵，rank(A)= r, 可以分解为：A=UDV',其中U'U=V'V=I。在R中可以用函数

scd()进行奇异值分解，例如：

> A=matrix(1:18,3,6)

> A

[,1] [,2] [,3] [,4] [,5] [,6]

[1,] 1 4 7 10 13 16

[2,] 2 5 8 11 14 17

[3,] 3 6 9 12 15 18

> svd(A)

$d

[1] 4.589453e+01 1.640705e+00 3.627301e-16

$u

[,1] [,2] [,3]

[1,] -0.5290354 0.74394551 0.4082483

[2,] -0.5760715 0.03840487 -0.8164966

[3,] -0.6231077 -0.66713577 0.4082483

$v

[,1] [,2] [,3]

[1,] -0.07736219 -0.7196003 -0.18918124

[2,] -0.19033085 -0.5089325 0.42405898

[3,] -0.30329950 -0.2982646 -0.45330031

[4,] -0.41626816 -0.0875968 -0.01637004

[5,] -0.52923682 0.1230711 0.64231130

[6,] -0.64220548 0.3337389 -0.40751869

> A.svd=svd(A)

> A.svd$u%*%diag(A.svd$d)%*%t(A.svd$v)

[,1] [,2] [,3] [,4] [,5] [,6]

[1,] 1 4 7 10 13 16

[2,] 2 5 8 11 14 17

[3,] 3 6 9 12 15 18

> t(A.svd$u)%*%A.svd$u

[,1] [,2] [,3]

[1,] 1.000000e+00 -1.169312e-16 -3.016793e-17

[2,] -1.169312e-16 1.000000e+00 -3.678156e-17

[3,] -3.016793e-17 -3.678156e-17 1.000000e+00

> t(A.svd$v)%*%A.svd$v

[,1] [,2] [,3]

[1,] 1.000000e+00 8.248068e-17 -3.903128e-18

[2,] 8.248068e-17 1.000000e+00 -2.103352e-17

[3,] -3.903128e-18 -2.103352e-17 1.000000e+00