6.54.5  Gram-Schmidt orthonormalization: gramschmidt

The gramschmidt command uses the Gram-Schmidt procedure to find an orthonormal set of vectors with the same span as a given set.

• gramschmidt takes one or two arguments.
  With one argument:
  • A, a matrix viewed as a list of row vectors or a list of elements that is a basis of a vector subspace.
  • f, a function that defines a scalar product on this vector space. If A is a matrix, then this is optional, and by default will be the standard scalar product.
• gramschmidt(A) or gramschmidt(L,f) returns an orthonormal basis for this scalar product.

Examples.

• Input:
  normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]]))
  or:
  normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]],dot))
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  √ 3 3 √ 3 3 √ 3 3 − √ 6 6 − √ 6 6 √ 6 3 − √ 2 2 √ 2 2 0

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

• Define a scalar product on the vector space of polynomials by:

  P· Q=

  ∫

  1 −1

  P(x) Q(x)dx

  Input:

  p_scal(p,q):=integrate(p*q,x,-1,1)

  gramschmidt([1,1+x],p_scal)

  or:
  gramschmidt([1,1+x],(p,q)->integrate(p*q,x,-1,1))
  Output:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

  1 √ 2

  ,

  1+x−1 √ 6 3

  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦