Gram-Schmidt Orthogonalization

Motivation

Assume we have a set of vectors ${| v_{i} ⟩}$ that are linearly independent, but not orthogonal to one another. Can we construct an orthormal set from them?

Procedure

Abstract

Normalize $| v_{1} ⟩ \to | 1 ⟩$
Take $|l_i⟩=|v_i⟩ - $ components of $| 1 ⟩, \dots, | i - 1 ⟩$
Normalize $| l_{i} ⟩$ put in $| i ⟩$ .

Concrete Procedure

$P_{n} = | n ⟩ ⟨ n |$ .

$| 1 ⟩ = \frac{| v_{1} ⟩}{| | | v_{1} ⟩ | |}$
$| l_{i} ⟩ = | v_{i} ⟩ - \sum_{j = 1}^{i - 1} P_{j} | v_{i} ⟩$
$| i ⟩ = \frac{| l_{i} ⟩}{| | | l_{i} ⟩ | |}$
Repeat 2-3 for $i = 2, \dots, n$ in increasing order
${| i ⟩}$ is a set of orthonormal basis vectors

Condensed

$| i ⟩ = \frac{| v_{i} ⟩ - \sum_{j = 1}^{i - 1} P_{j} | v_{i} ⟩}{| | | v_{i} ⟩ - \sum_{j = 1}^{i - 1} P_{j} | v_{i} ⟩ | |}$ .

Applications

AMO Physics: Perturbation Theory, Degenerate Eigenvalues
Accelerator Physics: Degeneracies
QR Decomposition
Legendre Polynomials ( ${1, x, x^{2}, \dots}$ )