Gram-Schmidt Orthonormalization

The orthonormalization algorithm proposed by Gram-Schmidt makes it possible to define the existence of orthonormal bases in a space and construct them (from any base).

From a set of vectors $ \vec{v_i} $ and its corresponding orthonormal basis, composed of the vectors $ \vec{e_i} $, then the Gram-Schmidt algorithm consists in calculating the orthogonal vectors $ \vec{u_i} $ which will allow to obtain the orthonormal vectors $ \vec{e_i} $ whose components are the following (the operator . is the scalar product on the vector space)

$$ \vec{u_1} = \vec{v_1} \ , \quad \vec{e_1} = \frac{ \vec{u_1} } { \| \vec{u_1} \| } $$

$$ \vec{u_2} = \vec{v_2} - \frac{ \vec{u_1} . \vec{v_2} }{ \vec{u_1} . \vec{u_1} } \vec{u_1} \ , \quad \vec{e_2} = \frac{ \vec{u_2} } { \| \vec{u_2} \| } $$

$$ \vec{u_3} = \vec{v_3} - \frac{ \vec{u_1} . \vec{v_3} }{ \vec{u_1} . \vec{u_1} } \vec{u_1} - \frac{ \vec{u_2} . \vec{v_3} }{ \vec{u_2} . \vec{u_2} } \vec{u_2} \ , \quad \vec{e_3} = \frac{ \vec{u_3} } { \| \vec{u_3} \| } $$

$$ \vec{u_k} = \vec{v_k} - \sum_{j=1}^{k-1} { \frac{ \vec{u_j} . \vec{v_k} }{ \vec{u_j} . \vec{u_j} } \vec{u_j} } \ , \quad \vec{e_k} = \frac{ \vec{u_k} } { \| \vec{u_k} \| } $$

Working with an orthonormal basis has many advantages. First of all, it makes it possible to simplify the calculations, because the coordinates of the vectors in this base are independent of each other. Moreover, it allows each vector in space to be represented in a unique way, which can be useful in many contexts.