Gram-Schmidt orthonormalisation

orthnorm() orthogonalises the columns of a matrix by the Gram-Schmidt process, optionally completes them to a basis of the whole space and scales them to unit length.

Usage

orthnorm(u = NULL, basis = TRUE, norm = TRUE)

# Default S3 method
orthnorm(u = NULL, basis = TRUE, norm = TRUE)

Arguments

u: a matrix (or vector) whose columns are orthogonalised.
basis: complete the columns to a basis of the whole space (default TRUE).
norm: scale the columns to unit length (default TRUE).

Value

The matrix of orthogonal (orthonormal with norm = TRUE) columns; NULL for u = NULL.

Details

For columns $\mathbf{u}_1, \ldots, \mathbf{u}_m$, $$\mathbf{v}_1 = \mathbf{u}_1, \qquad \mathbf{v}_i = \mathbf{u}_i - \sum_{h < i}\frac{\mathbf{u}_i^\top \mathbf{v}_h}{\mathbf{v}_h^\top\mathbf{v}_h}\mathbf{v}_h,$$ each divided by its length when norm = TRUE. With basis = TRUE and fewer columns than rows, unit vectors are added (those least explained by the given columns) to complete a basis.

References

Crossa, J., Perez-Elizalde, S., Jarquin, D., Cotes, J. M., Viele, K., Liu, G. and Cornelius, P. L. (2011). Bayesian estimation of the additive main effects and multiplicative interaction model. \emphCrop Science, \strong51(4), 1458–1469. \doi10.2135/cropsci2010.06.0343

Author

Muhammad Yaseen (myaseen208@gmail.com)

Examples

V <-
  orthnorm(
      u     = cbind(1, 1:4)
    , basis = TRUE
    , norm  = TRUE
    )
round(x = crossprod(V), digits = 10)
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#> [3,]    0    0    1    0
#> [4,]    0    0    0    1