BUILT-IN MATHEMATICA SYMBOL

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.

MORE INFORMATION

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.

Orthogonalize by default generates a Gram-Schmidt basis.

Other bases can be obtained by giving alternative settings for the Method option. Possible settings include: , , , and .

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:

Find the coefficients of a general vector with respect to this basis:

Find an orthonormal basis for two 3D vectors:

In[1]:=

Out[1]=

Find the coefficients of a general vector with respect to this basis:

In[2]:=

Out[2]=

Scope (2)

Use exact arithmetic to find an orthonormal basis:

Use machine arithmetic:

Use 25-digit precision arithmetic:

Orthogonalize complex vectors:

Generalizations & Extensions (2)

Find a symbolic basis, assuming all variables are real:

Orthogonalize symbolic expressions with a symbolic scalar product:

Options (3)

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:

Applications (1)

Derive normalized Legendre polynomials by orthogonalizing powers of

:

Derive normalized Hermite polynomials:

Properties & Relations (6)

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:

Orthogonalize[m] is related to QRDecomposition[Transpose[m]]:

They are the same up to sign: