This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
 Documentation / Mathematica / Add-ons / Standard Packages / LinearAlgebra  /

LinearAlgebra`Orthogonalization`

For most purposes, the most convenient type of basis for a vector space is orthonormal, i.e., the vectors are unit length and are pairwise orthogonal. The Gram-Schmidt procedure takes an arbitrary basis and generates an orthonormal one. It does this by sequentially processing the list of vectors, generating a vector perpendicular to the previous vectors in the list. For the process to succeed in producing an orthonormal set, the given vectors must be linearly independent. The function GramSchmidt assumes that this is the case. If the given vectors are not linearly independent, indeterminate or zero vectors may be produced.


Vector operations using the usual inner product.

  • This loads the package.
  • In[1]:= <<LinearAlgebra`Orthogonalization`

  • This applies the Gram-Schmidt procedure to the given list of three-dimensional vectors.
  • In[2]:= {w1, w2, w3} = GramSchmidt[
    {{3,4,2}, {2,5,2}, {1,2,6}}]

    Out[2]=

  • The result is an orthonormal basis, so the dot product of each pair of vectors is zero and each vector has unit length.
  • In[3]:= { w1 . w2, w2 . w3, w1 . w3,
    w1 . w1, w2 . w2, w3 . w3}

    Out[3]=

    A vector space is a generalization of the familiar notions of one-, two-, and three-dimensional space. In these familiar spaces we know how to compute lengths of vectors and the angle between two vectors. The usual way of doing this uses the dot product. The length of a vector is simply Sqrt[v.v] and two vectors are perpendicular (or orthogonal) if v.w is zero. In a more general vector space the dot product is replaced by an inner product. By setting the option InnerProduct, you can use GramSchmidt to produce a collection of vectors that are orthonormal with respect to your specified inner product.


    Vector operations using a specified inner product.

    The function you specify as the InnerProduct can be a pure function of two variables, with the variables given as #1 and #2. For example, the default InnerProduct->Dot could be expressed as InnerProduct->(#1.#2&). The parentheses around the formula for the function guarantee that Mathematica treats it as a single unit. For more information on pure functions, see The Mathematica Book.

  • The "vectors" here are functions, and the inner product is the integral of the product of two functions over the interval to

    .
  • In[4]:= GramSchmidt[{1, x, x^2, x^3, x^4},
    InnerProduct ->
    (Integrate[#1 #2,{x,-1,1}]&)] //Simplify

    Out[4]=

  • Here is the second Legendre polynomial normalized with respect to the same inner product. It is the same as the third element in the basis found using GramSchmidt.
  • In[5]:= Normalize[LegendreP[2,x],
    InnerProduct ->
    (Integrate[#1 #2,{x,-1,1}]&)]

    Out[5]=


    Generating orthogonal sets without normalizing.

  • The option Normalized is set to False.
  • In[6]:= {w1, w2} = GramSchmidt[{{3,4,3}, {2,3,6}},
    Normalized -> False]

    Out[6]=

  • The resulting vectors are orthogonal, but they are not normalized.
  • In[7]:= {w1 . w1, w1 . w2}

    Out[7]=