Orthogonalization: the Gram-Schmidt procedureVectors > Basics | Scalar product, Norms | Projection on a line | Orthogonalization | Hyperplanes | Linear functions | Application
A basis Example: An orthonormal basis in ![]() forms an orthonormal basis of What is orthogonalization?Orthogonalization refers to a procedure that finds an orthonormal basis of the span of given vectors. Given vectors ![]() where ![]() That is, the vectors Basic step: projection on a lineA basic step in the procedure consists in projecting a vector on a line passing through zero. Consider the line ![]() where The projection of a given point ![]() The vector ![]() Note that the vector ![]() where Gram-Schmidt procedureThe Gram-Schmidt procedure is a particular orthogonalization algorithm. The basic idea is to first orthogonalize each vector w.r.t. previous ones; then normalize result to have norm one. Case when the vectors are independentLet us assume that the vectors Gram-Schmidt procedure:
The GS process is well-defined, since at each step General case: the vectors may be dependentIt is possible to modify the algorithm to allow it to handle the case when the Modified Gram-Schmidt procedure:
On exit, the integer |