GramSchmidt: Gram-Schmidt Orthogonalization of a Matrix

In friendly/matlib: Matrix Functions for Teaching and Learning Linear Algebra and Multivariate Statistics

Gram-Schmidt Orthogonalization of a Matrix

Description

Carries out simple Gram-Schmidt orthogonalization of a matrix. Treating the columns of the matrix X in the given order, each successive column after the first is made orthogonal to all previous columns by subtracting their projections on the current column.

Usage

GramSchmidt(
  X,
  normalize = TRUE,
  verbose = FALSE,
  tol = sqrt(.Machine$double.eps),
  omit_zero_columns = TRUE
)

Arguments

X	a matrix
normalize	logical; should the resulting columns be normalized to unit length? The default is TRUE
verbose	logical; if TRUE, print intermediate steps. The default is FALSE
tol	the tolerance for detecting linear dependencies in the columns of a. The default is sqrt(.Machine$double.eps)
omit_zero_columns	if TRUE (the default), remove linearly dependent columns from the result

Value

A matrix of the same size as X, with orthogonal columns (but with 0 columns removed by default)

Author(s)

Phil Chalmers, John Fox

Examples

(xx <- matrix(c( 1:3, 3:1, 1, 0, -2), 3, 3))
crossprod(xx)
(zz <- GramSchmidt(xx, normalize=FALSE))
zapsmall(crossprod(zz))

# normalized
(zz <- GramSchmidt(xx))
zapsmall(crossprod(zz))

# print steps
GramSchmidt(xx, verbose=TRUE)

# A non-invertible matrix;  hence, it is of deficient rank
(xx <- matrix(c( 1:3, 3:1, 1, 0, -1), 3, 3))
R(xx)
crossprod(xx)
# GramSchmidt finds an orthonormal basis
(zz <- GramSchmidt(xx))
zapsmall(crossprod(zz))

friendly/matlib documentation built on March 3, 2024, 12:18 p.m.