QR computes the QR decomposition of a matrix, \(X\), that is an orthonormal matrix, \(Q\) and an upper triangular
matrix, \(R\), such that \(X = Q R\).
QR(X, tol = sqrt(.Machine$double.eps))a numeric matrix
tolerance for detecting linear dependencies in the columns of X
a list of three elements, consisting of an orthonormal matrix Q, an upper triangular matrix R, and the rank
of the matrix X
The QR decomposition plays an important role in many statistical techniques.
In particular it can be used to solve the equation \(Ax = b\) for given matrix \(A\) and vector \(b\).
The function is included here simply to show the algorithm of Gram-Schmidt orthogonalization. The standard
qr function is faster and more accurate.
A <- matrix(c(1,2,3,4,5,6,7,8,10), 3, 3) # a square nonsingular matrix
res <- QR(A)
res
#> $Q
#> [,1] [,2] [,3]
#> [1,] -0.2672612 -0.8728716 -0.4082483
#> [2,] -0.5345225 -0.2182179 0.8164966
#> [3,] -0.8017837 0.4364358 -0.4082483
#>
#> $R
#> [,1] [,2] [,3]
#> [1,] -3.741657 -8.552360 -14.1648458
#> [2,] 0.000000 -1.963961 -3.4914862
#> [3,] 0.000000 0.000000 -0.4082483
#>
#> $rank
#> [1] 3
#>
q <- res$Q
zapsmall( t(q) %*% q) # check that q' q = I
#> [,1] [,2] [,3]
#> [1,] 1 0 0
#> [2,] 0 1 0
#> [3,] 0 0 1
r <- res$R
q %*% r # check that q r = A
#> [,1] [,2] [,3]
#> [1,] 1 4 7
#> [2,] 2 5 8
#> [3,] 3 6 10
# relation to determinant: det(A) = prod(diag(R))
det(A)
#> [1] -3
prod(diag(r))
#> [1] -3
B <- matrix(1:9, 3, 3) # a singular matrix
QR(B)
#> $Q
#> [,1] [,2] [,3]
#> [1,] -0.2672612 -0.8728716 0
#> [2,] -0.5345225 -0.2182179 0
#> [3,] -0.8017837 0.4364358 0
#>
#> $R
#> [,1] [,2] [,3]
#> [1,] -3.741657 -8.552360 -13.363062
#> [2,] 0.000000 -1.963961 -3.927922
#> [3,] 0.000000 0.000000 0.000000
#>
#> $rank
#> [1] 2
#>