Gram-Schmidt orthonormalization

### 15.6.5 Gram-Schmidt orthonormalization

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:
  - 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

normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]]))

or:

normal(gramschmidt([[1,1,1],[0,0,1],[0,1,0]],dot))

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
√ |
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=∫−11P(x) Q(x) dx. Then:

p_scal(p,q):=integrate(p*q,x,-1,1):; gramschmidt([1,1+x],p_scal)

| ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
1
√ |
2
| , |
1+x−1
√ |
6
3
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦