R语言矩阵奇异值分解实例svd(A)

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矩阵奇异值分解
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