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,]  0.255317055  0.6215527 -0.2473253
#> [2,] -2.437263611  1.1484116 -0.2441996
#> [3,] -0.005571287 -1.8218177 -0.2827054
GramSchmidt(A[1, ], A[2, ], A[3, ])
#>           [,1]       [,2]        [,3]
#> v1  0.35657819  0.8680664 -0.34541684
#> v2 -0.93175210  0.3575222 -0.06337134
#> v3 -0.06848364 -0.3444397 -0.93630726

# 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,] -0.5536994  0.628982 0.0000000
#> [2,]  1.0000000  0.000000 0.0000000
#> [3,]  2.0650249 -1.630989 0.5124269
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