A為m×n矩陣可以進(jìn)行QR分解，A=QR，其中：Q'Q＝I，在R中可以用函數(shù)qr()進(jìn)行

QR分解，例如：

> A=matrix(1:16,4,4)

> qr(A)

$qr

[,1] [,2] [,3] [,4]

[1,] -5.4772256 -12.7801930 -2.008316e+01 -2.738613e+01

[2,] 0.3651484 -3.2659863 -6.531973e+00 -9.797959e+00

[3,] 0.5477226 -0.3781696 2.641083e-15 2.056562e-15

[4,] 0.7302967 -0.9124744 8.583032e-01 -2.111449e-16

$rank

[1] 2

$qraux

[1] 1.182574e+00 1.156135e+00 1.513143e+00 2.111449e-16

$pivot

[1] 1 2 3 4

attr(,"class")

[1] "qr"

rank項(xiàng)返回矩陣的秩，qr項(xiàng)包含了矩陣Q和R的信息，要得到矩陣Q和R，可以用函數(shù)

qr.Q()和qr.R()作用qr()的返回結(jié)果，例如：

> qr.R(qr(A))

[,1] [,2] [,3] [,4]

[1,] -5.477226 -12.780193 -2.008316e+01 -2.738613e+01

[2,] 0.000000 -3.265986 -6.531973e+00 -9.797959e+00

[3,] 0.000000 0.000000 2.641083e-15 2.056562e-15

[4,] 0.000000 0.000000 0.000000e+00 -2.111449e-16

> qr.Q(qr(A))

[,1] [,2] [,3] [,4]

[1,] -0.1825742 -8.164966e-01 -0.4000874 -0.37407225

[2,] -0.3651484 -4.082483e-01 0.2546329 0.79697056

[3,] -0.5477226 -8.131516e-19 0.6909965 -0.47172438

[4,] -0.7302967 4.082483e-01 -0.5455419 0.04882607

> qr.Q(qr(A))%*%qr.R(qr(A))

[,1] [,2] [,3] [,4]

[1,] 1 5 9 13

[2,] 2 6 10 14

[3,] 3 7 11 15

[4,] 4 8 12 16

> t(qr.Q(qr(A)))%*%qr.Q(qr(A))

[,1] [,2] [,3] [,4]

[1,] 1.000000e+00 -1.457168e-16 -6.760001e-17 -7.659550e-17

[2,] -1.457168e-16 1.000000e+00 -4.269046e-17 7.011739e-17

[3,] -6.760001e-17 -4.269046e-17 1.000000e+00 -1.596437e-16

[4,] -7.659550e-17 7.011739e-17 -1.596437e-16 1.000000e+00

> qr.X(qr(A))

[,1] [,2] [,3] [,4]

[1,] 1 5 9 13

[2,] 2 6 10 14

[3,] 3 7 11 15

[4,] 4 8 12 16