Definition

Method for orthogonalizing a set of vectors.

Computation

Given $a_{1}, a_{2}, \dots, a_{n + 1} \in R^{n}$

$a_{1}$ : $A_{1} := a_{1}, q_{1} = \frac{A _{1}}{∣∣ A _{1} ∣∣}$
$a_{2}$ : $A_{2} := (a_{2} - (q_{1}^{⊺} a_{2}) q_{1}) ⊥ q_{1}, q_{2} = \frac{A _{2}}{∣∣ A _{2} ∣∣}$
$a_{3}$ : $A_{3} := (a_{3} - (q_{1}^{⊺} a_{3}) q_{1} - (q_{2}^{⊺} a_{3}) q_{2}) ⊥ q_{1}, q_{2}, q_{3} = \frac{A _{3}}{∣∣ A _{3} ∣∣}$
$a_{k}$ : $A_{k} := (a_{k} - \sum_{i = 1}^{k - 1} (q_{i}^{⊺} a_{k}) q_{i}) ⊥ q_{1}, q_{2}, \dots, q_{k - 1}, q_{k} = \frac{A _{k}}{∣∣ A _{k} ∣∣}$ Where $(q_{i}^{⊺} a_{k}) q_{i}$ means the projection of the vector $a_{k}$ onto the vector space of $A_{i}$ .

Therefore, it can be expressed using $proj_{A_{i}} (a_{k})$

$a_{1}$ : $A_{1} := a_{1}, q_{1} = \frac{A _{1}}{∣∣ A _{1} ∣∣}$
$a_{2}$ : $A_{2} := (a_{2} - proj_{A_{1}} (a_{2})) ⊥ q_{1}, q_{2} = \frac{A _{2}}{∣∣ A _{2} ∣∣}$
$a_{3}$ : $A_{3} := (a_{3} - proj_{A_{1}} (a_{3}) - proj_{A_{2}} (a_{3})) ⊥ q_{1}, q_{2}, q_{3} = \frac{A _{3}}{∣∣ A _{3} ∣∣}$
$a_{k}$ : $A_{k} := (a_{k} - \sum_{i = 1}^{k - 1} proj_{A_{i}} (a_{k})) ⊥ q_{1}, q_{2}, \dots, q_{k - 1}, q_{k} = \frac{A _{k}}{∣∣ A _{k} ∣∣}$