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.

Duncan Murdoch

## Examples

# Proceed through the rows in order
print(A <- matrix(rnorm(9), 3, 3))
#>            [,1]       [,2]       [,3]
#> [1,] -0.8074924  0.1737682 -1.5408661
#> [2,]  0.9076540  0.0719223  1.2413485
#> [3,]  0.7820587 -0.2130351 -0.6859608
GramSchmidt(A[1, ], A[2, ], A[3, ])
#>          [,1]        [,2]       [,3]
#> v1 -0.4618763  0.09939337 -0.8813576
#> v2  0.6655128  0.69572155 -0.2703040
#> v3  0.5863131 -0.71140178 -0.3874849

# 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,]  1.1137372 -1.0624995  0.0000000
#> [2,]  1.0000000  0.0000000  0.0000000
#> [3,] -0.2092996 -0.6374341 -0.8528405
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