Orthogonalize

Orthogonalize[{v1,v2,}]

gives an orthonormal basis found by orthogonalizing the vectors vi.

Orthogonalize[{e1,e2,},f]

gives a basis for the ei orthonormal with respect to the inner product function f.

Details and Options

  • Orthogonalize[{v1,v2,}] 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 ei.
  • The ei can be any expressions for which f always yields real results.
  • Orthogonalize[{v1,v2,},Dot] effectively assumes that all elements of the vi are real.
  • Orthogonalize by default generates a GramSchmidt basis.
  • Other bases can be obtained by giving alternative settings for the Method option. Possible settings include: "GramSchmidt", "ModifiedGramSchmidt", "Reorthogonalization", and "Householder".
  • Orthogonalize[list,Tolerance->t] sets to zero elements whose relative norm falls below t.

Examples

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Basic Examples  (1)

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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Scope  (2)

Generalizations & Extensions  (2)

Options  (3)

Applications  (1)

Properties & Relations  (6)

See Also

OrthogonalMatrixQ  UnitaryMatrixQ  Projection  Normalize  Dot  Inner  QRDecomposition  LinearSolve

Tutorials

Introduced in 2007
(6.0)