Gram-Schmidt OrthogonalizationGram-Schmidt orthogonalization generates an orthonormal basis from an arbitrary basis. An orthonormal basis is a basis,  , for which In Mathematica a Gram-Schmidt orthogonalization can be computed from a set of vectors with the package function GramSchmidt, which is defined in the package LinearAlgebra`Orthogonalization`. This loads the package. In[1]:=  |
This creates a set of three vectors that form a basis for  . In[2]:=  |
A plot visualizes the vectors; they all tend to lie in the same direction. In[5]:=  |
This computes an orthonormal basis. In[6]:=  |
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The orthonormal vectors are obviously much more spread out. In[7]:=  |
The vectors v1, v2, and v3 are orthonormal, thus the dot product of each vector with itself is 1. In[8]:=  |
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In addition, the dot product of a vector with another vector is 0. In[9]:=  |
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This uses Outer to compare all vectors with all other vectors. In[10]:=  |
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