Gram-Schmidt Orthogonalization

Motivation

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

Procedure

Abstract

Normalize $|v_1\rangle \to |1\rangle$
Take $|l_i⟩=|v_i⟩ - $ components of $|1\rangle,\cdots,|i-1\rangle$
Normalize $|l_i\rangle$ put in $|i\rangle$.

Concrete Procedure

$P_n = |n\rangle\langle n|$.

$|1\rangle = \frac{|v_1\rangle}{|||v_1\rangle||}$
$|l_i\rangle = |v_i\rangle - \sum_{j=1}^{i-1} P_j|v_i\rangle$
$|i\rangle = \frac{|l_i\rangle}{|||l_i\rangle||}$
Repeat 2-3 for $i=2,\cdots,n$ in increasing order
$\{|i\rangle\}$ is a set of orthonormal basis vectors

Condensed

$|i\rangle = \frac{|v_i\rangle - \sum_{j=1}^{i-1}P_j|v_i\rangle}{|||v_i\rangle - \sum_{j=1}^{i-1}P_j|v_i\rangle||}$.

Applications

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