proof of Gram-Schmidt orthogonalization procedure
Note that, while we state the following as a theorem for the sake of logical completeness and to establish notation, our definition of Gram-Schmidt orthogonalization is wholly equivalent to that given in the defining entry.
Theorem.
(Gram-Schmidt Orthogonalization) Let be a basis for an inner product space with inner product . Define and recursively by
(1) |
where for . Then is an orthonormal basis for .
Proof.
We proceed by induction on . In the case , we suppose is a basis for the inner product space . Letting , it is clear that , whence it follows that . Thus is an orthonormal basis for , and the result holds for . Now let , and suppose the result holds for arbitrary . Let be a basis for an inner product space . By the inductive hypothesis we may use to construct an orthonormal set of vectors such that . In accordance with the procedure outlined in the statement of the theorem, let be defined as
First we show that the vectors are mutually orthogonal. Consider the inner product of with for . By construction, we have
Now since is an orthonormal set of vectors, whence , each term in the summation on the right-hand side of the preceding equation will vanish except for the term where . Thus by this and the preceding equation, we have
Thus is orthogonal to for , so we may take to have an orthonormal set of vectors. Finally we show that is a basis for . By construction, each is a linear combination of the vectors , so we have orthogonal, hence linearly independent vectors in the dimensional space , from which it follows that is a basis for . Thus the result holds for , and by the principle of induction, for all . ∎
Title | proof of Gram-Schmidt orthogonalization procedure |
Canonical name | ProofOfGramSchmidtOrthogonalizationProcedure |
Date of creation | 2013-03-22 16:27:17 |
Last modified on | 2013-03-22 16:27:17 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 11 |
Author | rspuzio (6075) |
Entry type | Proof |
Classification | msc 65F25 |
Synonym | Gram-Schmidt |
Synonym | orthogonalization |
Related topic | GramSchmidtOrthogonalization |
Related topic | ExampleOfGramSchmidtOrthogonalization |
Related topic | InnerProductSpace |
Related topic | InnerProduct |
Related topic | QRDecomposition |
Related topic | NormedVectorSpace |
Related topic | Orthogonal |
Related topic | OrthogonalVectors |
Related topic | Basis |
Related topic | Span |
Related topic | LinearIndependence |
Related topic | Orthonormal |
Related topic | OrthonormalBasis |
Defines | Gram-Schmidt orthogonalization |