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,] -1.4000435 -0.005571287 -1.8218177
#> [2,] 0.2553171 0.621552721 -0.2473253
#> [3,] -2.4372636 1.148411606 -0.2441996
GramSchmidt(A[1, ], A[2, ], A[3, ])
#> [,1] [,2] [,3]
#> v1 -0.6093385 -0.002424781 -0.7929065
#> v2 0.3903683 0.869490850 -0.3026520
#> v3 -0.6901588 0.493943096 0.5288677
# 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,] -0.2827054 -0.5536994 0.000000
#> [2,] 1.0000000 0.0000000 0.000000
#> [3,] 0.6289820 2.0650249 -1.630989
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