Matrix Action 
Perpframes, Aligners and Hangers 
Matrix Subspaces  
Linear Systems, Pseudo-Inverse 
Condition Number 
Matrix Norm, Rank One 
Data Compression 
Noise Filtering 
Todd Will
UW-La Crosse
More Stretchers 


Changing Dimensions 


Look at the action of [Graphics:stretchergr1.gif]

When you look at that action you can see why it's natural to call a diagonal matrix a "stretcher" matrix. 

The diagonal matrix [Graphics:stretchergr6.gif] stretches in the x direction by a factor of "a" and in the y direction by a factor of "b". 

You can verify this by hand using the column way to multiply a matrix times a vector: 


Check out a few more stretchers.


Stretching by 1/2 squashes the circle in the y direction. 

Stretching by 0 in the y direction squashes the circle onto the x-axis. 

Stretching by -2 in the x-direction, means flipping across the y-axis as well as stretching. 

Changing dimensions

Both [Graphics:stretchergr21.gif]and [Graphics:stretchergr22.gif]are stretcher matrices since their non-diagonal entries are zero. 

The matrix [Graphics:stretchergr23.gif] sends [Graphics:stretchergr24.gif]to [Graphics:stretchergr25.gif]

But[Graphics:stretchergr26.gif]sends [Graphics:stretchergr27.gif]to [Graphics:stretchergr28.gif]

Check out the action of each of these stretchers. 

A stretch by a factor of 5 in the x direction and a factor of 2 in the y direction. 

A stretch by a factor of 5 in the x direction and a factor of 2 in the y direction. 

But note how the stretcher matrix [Graphics:stretchergr34.gif]not only stretches the 2D circle but also embeds the ellipse into 3 dimensional space. 


1.  Check out the following ellipse. 

You can get this ellipse by stretching the unit circle by a factor of 3 in the x direction and a factor of 2 in the y direction. 

To get the ellipse shown above I would hit the unit circle with (choose one): 

(a) the matrix [Graphics:stretchergr37.gif] 

(b) the matrix [Graphics:stretchergr38.gif] 

(c) the matrix [Graphics:stretchergr39.gif] 

2.  Check out the following ellipsoid 


You can get this ellipsoid by stretching the unit sphere by 

  • a factor of 4 in the x direction 
  • a factor of 8 in the y direction
  • a factor of 2 in the z direction
To get the ellipsoid shown above I would hit the unit sphere with (choose one): 

(a) the matrix [Graphics:stretchergr42.gif] 

(b) the matrix [Graphics:stretchergr43.gif] 

(c) the matrix [Graphics:stretchergr44.gif] 

© 1999 Todd Will
Last Modified: 07-Jan-1999
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