GramSchmidt | R Documentation |
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.
GramSchmidt( X, normalize = TRUE, verbose = FALSE, tol = sqrt(.Machine$double.eps) )
X |
a matrix |
normalize |
logical; should the resulting columns be normalized to unit length? |
verbose |
logical; if |
tol |
the tolerance for detecting linear dependencies in the columns of a. The default is |
A matrix of the same size as X
, with orthogonal columns
Phil Chalmers, John Fox
(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))
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