Description:
Gram-Schmidt orthogonalization is a way of converting a given arbitrary basis into an equivalent orthonormal basis:
This often leads to better accuracy (e.g. in least square problems) and/or simplifications.
Modified Gram-Schmidt Procedure
Given a set of linearly independent vectors, ,turn them into an equivalent orthonormal set as follows:
Step 1:
Note that is the component of in the direction of :
Also .The matrix is called the projection operator onto .Ignore in the remaining process.
Step 2:
Repeat the process along the same lines until you run out of vectors.
Graphical example:
Normalize : Eliminate the components in the direction from the rest: Normalize : Eliminate the components in the direction from the rest: Normalize :