Gram-Schmidt

Description:

Gram-Schmidt (or Gramm-Schmidt?) 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

Start with the first vector, putting i=1. Now

1.

Normalize vector :

2.

For the remaining vectors , eliminate their component in the directions of using the following formula:

3.

Increment i and repeat until there are no vectors left.

Note that for real vectors

is indeed the component of in the direction of :

Graphical example:

Normalize :

Eliminate the components in the direction from the rest:

Normalize :

Eliminate the components in the direction from the rest:

Normalize :

Projection:

The component substraction formula can also be read as

The matrix is the projection operator onto 10/11/00 0:32:32 10/11/00 0:35:18 10/13/00 0:05:06