The Gram-Schmidt algorithm

Generate a 3x3 orthogonal matrix using the Gram-Schmidt algorithm.

GramSchmidt(v1, v2, v3, order = 1:3)

Arguments

v1, v2, v3	Three length 3 vectors (taken as row vectors).
order	The precedence order for the vectors; see Details.

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.

Value

A 3x3 matrix whose rows are the orthogonalization of the original row vectors.

Author

Duncan Murdoch

Examples

# Proceed through the rows in order
print(A <- matrix(rnorm(9), 3, 3))
#>             [,1]       [,2]       [,3]
#> [1,]  1.06730788 -0.0499649 2.75541758
#> [2,]  0.07003485 -0.2514834 0.04653138
#> [3,] -0.63912332  0.4447971 0.57770907
GramSchmidt(A[1, ], A[2, ], A[3, ])
#>          [,1]        [,2]        [,3]
#> v1  0.3611469 -0.01690671  0.93235566
#> v2  0.1713997 -0.98159768 -0.08419111
#> v3 -0.9166215 -0.19021082  0.35160316

# 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,] 0.1181949 -1.9117205  0.0000000
#> [2,] 1.0000000  0.0000000  0.0000000
#> [3,] 0.8620865 -0.2432367 -0.2060872
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