Gram-Schmidt

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:

1.

Normalize the first vector . That will be your

2.

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

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:

1.

Normalize the second vector . That will be your

2.

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

Ignore in the remaining process.

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 :