matlib (version 1.0.0)

QR: QR Decomposition by Graham-Schmidt Orthonormalization

Description

QR computes the QR decomposition of a matrix, \(X\), that is an orthonormal matrix, \(Q\) and an upper triangular matrix, \(R\), such that \(X = Q R\).

Usage

QR(X, tol = sqrt(.Machine$double.eps))

Value

a list of three elements, consisting of an orthonormal matrix Q, an upper triangular matrix R, and the rank

of the matrix X

Arguments

X

a numeric matrix

tol

tolerance for detecting linear dependencies in the columns of X

Author

John Fox and Georges Monette

Details

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.

See Also

qr

Examples

A <- matrix(c(1,2,3,4,5,6,7,8,10), 3, 3) # a square nonsingular matrix
res <- QR(A)
res
q <- res$Q
zapsmall( t(q) %*% q)   # check that q' q = I
r <- res$R
q %*% r                 # check that q r = A

# relation to determinant: det(A) = prod(diag(R))
det(A)
prod(diag(r))

B <- matrix(1:9, 3, 3) # a singular matrix
QR(B)