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Orthogonalize

Orthogonalize
gives an orthonormal basis found by orthogonalizing the vectors .
Orthogonalize
gives a basis for the orthonormal with respect to the inner product function f.
  • Orthogonalize uses the ordinary scalar product as an inner product.
  • The output from Orthogonalize always contains the same number of vectors as the input. If some of the input vectors are not linearly independent, the output will contain zero vectors.
  • All nonzero vectors in the output are normalized to unit length.
  • The inner product function f is applied to pairs of linear combinations of the .
  • The can be any expressions for which f always yields real results.
  • Orthogonalize[{v1, v2, ...}, Dot] effectively assumes that all elements of the are real.
  • Other bases can be obtained by giving alternative settings for the Method option. Possible settings include: , , , and .
Find an orthonormal basis for two 3D vectors:
Find the coefficients of a general vector with respect to this basis:
Find an orthonormal basis for two 3D vectors:
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Find the coefficients of a general vector with respect to this basis:
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Use exact arithmetic to find an orthonormal basis:
Use machine arithmetic:
Use 25-digit precision arithmetic:
Orthogonalize complex vectors:
Find a symbolic basis, assuming all variables are real:
Orthogonalize symbolic expressions with a symbolic scalar product:
Below the tolerance, two vectors are not recognized as linearly independent:
forms a set of vectors that are nearly linearly dependent:
Deviation from orthonormality for the default method:
Deviation for all of the methods:
For a large numerical matrix, the Householder method is usually fastest:
Derive normalized Legendre polynomials by orthogonalizing powers of :
Derive normalized Hermite polynomials:
In dimensions, there can be at most elements in the orthonormal basis:
Most sets of random -dimensional vectors are spanned by exactly basis vectors:
With the default method, the first element of the basis is always a multiple of the first vector:
For linearly independent vectors, the result is an orthonormal set:
Verify using matrix multiplication:
For linearly independent vectors, the result is a set orthonormal with the given inner product:
Verify orthonormality:
They are the same up to sign:
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