Gram-Schmidt Orthonormalization

Gram-Schmidt orthonormalization, also called the Gram-Schmidt process, constructs an orthonormal basis for the span of a linearly independent family in an inner product space, preserving the successive spans. Omitting the normalization steps gives Gram-Schmidt orthogonalization.

For vectors , , ... in a real or complex inner product space, the construction is

where and the sum is empty for . Here denotes the given inner product, taken to be linear in its first argument and antilinear in its second, and is its induced norm. The linear independence of the input vectors ensures that every residual is nonzero, so the normalization is defined (Axler 2024, pp. 183-184 and 201). In the Wolfram Language, Orthogonalize[v0, v1, ...] gives an orthonormal basis for a linearly independent list of vectors.

For the real functions considered below, use the weighted inner product

The nonnegative weight must make this inner product positive definite on the function space under consideration, and all required integrals must be finite. All subsequent integrals are over the same interval .

Applying the Gram-Schmidt process to the functions 1, , , ... on the interval with the usual inner product gives the Legendre polynomials (up to constant multiples; Reed and Simon 1972, p. 47).

Given an original set of linearly independent functions , let denote the orthogonalized (but not normalized) functions, denote the orthonormalized functions, and define

Then take

where we require

By definition,

so

The first orthogonalized function is therefore

and the corresponding normalized function is

By mathematical induction, it follows that

where

and

If the functions are normalized to positive instead of 1, then

Orthogonal polynomials are especially easy to generate using the Gram-Schmidt process applied to 1, , , ..., provided all required moments are finite. In particular,

Choose monic polynomials of degree , starting with

As defined, and are orthogonal polynomials, as can be seen from

Now use the recurrence relation

for to construct the remaining monic polynomials.

To verify orthogonality, suppose , ..., are mutually orthogonal, and write the two coefficients in the recurrence relation as and . Then

For , all three terms vanish because has degree at most . For , the polynomial also has degree at most , so

This cancels the term containing . For , the first two terms cancel by the definition of , and the last term is zero. Thus is orthogonal to every preceding polynomial, completing the induction.

Many common orthogonal polynomials of mathematical physics can be generated in this manner. In finite-precision arithmetic, the classical process can lose orthogonality (Golub and Van Loan 1996).