# Connexions

You are here: Home » Content » Matrix Analysis » Gram-Schmidt Orthogonalization

### Lenses

What is a lens?

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

#### Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Rice Digital Scholarship

This collection is included in aLens by: Digital Scholarship at Rice University

Click the "Rice Digital Scholarship" link to see all content affiliated with them.

#### Also in these lenses

• Lens for Engineering

This module and collection are included inLens: Lens for Engineering
By: Sidney Burrus

Click the "Lens for Engineering" link to see all content selected in this lens.

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Steven J. Cox. E-mail the author

# Gram-Schmidt Orthogonalization

Module by: Steven J. Cox. E-mail the author

Summary: (Blank Abstract)

Suppose that MM is an mm-dimensional subspace with basis x1xm x1 xm We transform this into an orthonormal basis q1qm q1 qm for MM via

1. Set y1=x1 y1 x1 and q1=y1y1 q1 y1 y1
2. y2=x2 y2 x2 minus the projection of x2x2 onto the line spanned by q1q1. That is y2=x2q1q1Tq1-1q1Tx2=x2q1q1Tx2 y2 x2 q1 q1 q1 q1 x2 x2 q1 q1 x2 Set q2=y2y2 q2 y2 y2 and Q2= q1 q2 Q2 q1 q2 .
3. y3=x3 y3 x3 minus the projection of x3x3 onto the plane spanned by q1q1 and q2q2. That is y3=x3Q2Q2TQ2-1Q2Tx3=x3q1q1Tx3 y3 x3 Q2 Q2 Q2 Q2 x3 x3 q1 q1 x3 q2 q2 x3 Set q3=y3y3 q3 y3 y3 and Q3=q1q2q3 Q3 q1 q2 q3 . Continue in this fashion through step (mm).
• (mm) ym=xm ym xm minus its projection onto the subspace spanned by the columns of Q m 1 Q m 1 . That is ym=xm Q m 1 Q m 1 T Q m 1 -1 Q m 1 Txm xmj=1m1qjqjTxm ym xm Q m 1 Q m 1 Q m 1 Q m 1 xm xm j 1 m 1 qj qj xm
Set qm=ymym qm ym ym To take a simple example, let us orthogonalize the following basis for R3 3 , x1=( 1 0 0 ) x2=( 1 1 0 ) x3=( 1 1 1 ) x1 1 0 0 x2 1 1 0 x3 1 1 1
1. q1=y1=x1 q1 y1 x1 .
2. y2=x2q1q1Tx2=( 010 )T y2 x2 q1 q1 x2 0 1 0 and so, q2=y2 q2 y2 .
3. y3=x3q1q1Tx3=( 001 )T y3 x3 q1 q1 x3 q2 q2 x3 0 0 1 and so, q3=y3 q3 y3 .
We have arrived at q1=( 1 0 0 ) q2=( 0 1 0 ) q3=( 0 0 1 ) q1 1 0 0 q2 0 1 0 q3 0 0 1 . Once the idea is grasped the actual calculations are best left to a machine. Matlab accomplishes this via the orth command. Its implementation is a bit more sophisticated than a blind run through our steps (1) through (mm). As a result, there is no guarantee that it will return the same basis. For example


>>X=[1 1 1;0 1 1 ;0 0 1];

>>Q=orth(X)

Q=

0.7370  -0.5910  0.3280

0.5910   0.3280 -0.7370

0.3280   0.7370  0.5910



This ambiguity does not bother us, for one orthogonal basis is as good as another. Let us put this into practice, via (10.8).

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks