SVD

Matrix Norm
Rank One

Image Compression

Noise Filtering

Todd Will
UW-La Crosse

Matrix Hits

2D Action

Matrix Action

3D Action

Exercises

Matrix Hits

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

When the matrix hits you get .

Two different ways to compute a matrix times a point.

COLUMN WAY to compute

: Take a linear combination of the COLUMNS of A using the weights from

. E.g.

ROW WAY to compute : Dot each ROW of A with .
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

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.

You can hit other curves too.

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

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.

Exercises

The plot to the right shows a matrix A hitting the unit circle.

Give estimates for A and A.
Give estimates for two different points that A sends to {0,0}.

Todd Will UW-La Crosse

Matrix Action

Matrix Hits

Two different ways to compute a matrix times a point.

2D Matrix Action

Check out a few more.

You can hit other curves too.

3D Matrix Action

Exercises

Todd Will
UW-La Crosse