GramSchmidt.RdGenerate a 3x3 orthogonal matrix using the Gram-Schmidt algorithm.
GramSchmidt(v1, v2, v3, order = 1:3)Three length 3 vectors (taken as row vectors).
The precedence order for the vectors; see Details.
This function orthogonalizes the matrix rbind(v1, v2, v3)
using the Gram-Schmidt algorithm. It can handle rank 2 matrices
(returning a rank 3 matrix). If the original is rank 1, it is likely
to fail.
The order vector determines the precedence of the original
vectors. For example, if it is c(i, j, k), then row i
will be unchanged (other than normalization); row j will
normally be transformed within the span of rows i and j.
Row k will be transformed orthogonally to the span of
the others.
A 3x3 matrix whose rows are the orthogonalization of the original row vectors.
# Proceed through the rows in order
print(A <- matrix(rnorm(9), 3, 3))
#> [,1] [,2] [,3]
#> [1,] -0.4940813 -0.647212 3.491336e-01
#> [2,] 1.5566400 -1.126534 9.691385e-01
#> [3,] -0.9526286 -1.350823 5.079785e-05
GramSchmidt(A[1, ], A[2, ], A[3, ])
#> [,1] [,2] [,3]
#> v1 -0.5576901 -0.7305350 0.3940816
#> v2 0.8207012 -0.4142486 0.3935068
#> v3 0.1242227 -0.5428781 -0.8305733
# Keep the middle row unchanged
print(A <- matrix(c(rnorm(2), 0, 1, 0, 0, rnorm(3)), 3, 3, byrow = TRUE))
#> [,1] [,2] [,3]
#> [1,] -0.1377840 -0.6537328 0.0000000
#> [2,] 1.0000000 0.0000000 0.0000000
#> [3,] -0.4113343 -0.2977269 0.6723348
GramSchmidt(A[1, ], A[2, ], A[3, ], order = c(2, 1, 3))
#> [,1] [,2] [,3]
#> v2 0 -1 0
#> v1 1 0 0
#> v3 0 0 1