Gram-Schmidt on a subspace of vectors

gram-schmidt.r

cols <- c(
  c(3, 9, 2, 3, 2),
  c(8, 8, 1, 9, 0),
  c(1, 9, 5, 9, 8),
  c(8, 10, 7, 5, 3),
  c(5, 8, 8, 0, 8),
  c(5, 10, 4, 7, 2))

# prepare column vectors matrix
mat <- matrix(cols, byrow = FALSE, nrow = 5)

# prepare orthogonal complement matrix
complement_matrix <- matrix(mat[, 1], ncol = 1)

# define projection function
project <- function(from, onto) {
  nom <- t(from) %*% onto
  denom <- t(onto) %*% onto
  return ((nom / denom) %*% onto)
}

# loop through each column starting from 2
for (i in 2:ncol(mat)) {
  v <- mat[, i]
  u <- v
  for (j in 1:(i-1)) {
    u <- u - project(v, complement_matrix[, j])
  }
  
  # append new column to our matrix
  complement_matrix <- cbind(complement_matrix, t(u))
}

# show our complement matrix
print(complement_matrix)

# we now have the orthogonal complement matrix, lets make it orthonormal
normalize_vector <- function(vector) {
  return (vector / sqrt(vector %*% vector))
}

orthonormal_matrix <- apply(complement_matrix, 2, normalize_vector)

# show our orthonormal_matrix
print(orthonormal_matrix)

# prove that 2 vectors are orthogonal:
print(orthonormal_matrix[, 2] %*% orthonormal_matrix[, 4])

# prove that any vector is unit length
print(sqrt(orthonormal_matrix[, 2] %*% orthonormal_matrix[, 2]))