# 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

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

### Concrete Procedure

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

1. $$|1\rangle = \frac{|v_1\rangle}{|||v_1\rangle||}$$
2. $$|l_i\rangle = |v_i\rangle - \sum_{j=1}^{i-1} P_j|v_i\rangle$$
3. $$|i\rangle = \frac{|l_i\rangle}{|||l_i\rangle||}$$
4. Repeat 2-3 for $$i=2,\cdots,n$$ in increasing order
5. $$\{|i\rangle\}$$ 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 ($$\{1,x,x^2,\cdots\}$$)

Created: 2024-05-30 Thu 21:15

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