Matrix Action 
Aligners and Hangers 
Matrix Subspaces  
Linear Systems 
Condition Number 
Matrix Norm 
Rank One 
Image Compression 
Noise Filtering 
Todd Will
UW-La Crosse
 Matrix Hits 
2D Action

Matrix Action

3D Action 

Matrix Hits

When you hit a point with a matrix you get another point. 

When the matrix [Graphics:actiongr1.gif] hits [Graphics:actiongr2.gif] you get [Graphics:actiongr3.gif]

Two different ways to compute a matrix times a point. 

COLUMN WAY to compute [Graphics:actiongr4.gif]: Take a linear combination of the COLUMNS of A using the weights from [Graphics:actiongr5.gif]. E.g. [Graphics:actiongr6.gif] 

ROW WAY to compute [Graphics:actiongr7.gif]: Dot each ROW of A with [Graphics:actiongr8.gif]
E.g. [Graphics:actiongr9.gif] 

2D Matrix Action 

The plot below shows some color coded points on the unit circle. Move your mouse over the plot to see what happens when the matrix [Graphics:actiongr10.gif]hits the points on the unit circle. 
  You can see that the matrix stretches and rotates the unit circle--that's Matrix Action. 

Check out a few more. 

Put your cursor over a matrix to see the matrix hit the unit circle. 

[Graphics:matgr1.gif] [Graphics:matgr3.gif] [Graphics:matgr4.gif]


You can hit other curves too.  

Check out the same three matrices hitting a damped sine wave. 

[Graphics:matgr1.gif]  [Graphics:matgr3.gif]  [Graphics:matgr4.gif] 

3D Matrix Action

If you have a 3 by 3 matrix you can hit 3D surfaces. 

Check out the matrix [Graphics:matgr5.gif] hitting this surface. 
  The matrix appears to be rotating and stretching the surface. That's 3D matrix action.


The plot to the right shows a matrix A hitting the unit circle. 
  1. Give estimates for A and A.
  2. Give estimates for two different points that A sends to {0,0}.

© 1999 Todd Will
Last Modified: 03-Jan-99
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