for another 
matrix 
.Singular Values
Here's the picture of for another matrix .
The plot on the left shows a grid spanned by an orthonormal basis 
.
The matrix 
sends this grid to an orthogonal grid on the right spanned by 
.
The grid on the left has 
squares, but the grid on the right has 
rectangles.
To make the grid on the right have the same scale as the one on the left add unit vectors ![FormBox[RowBox[{{v_1, v_2}, Cell[]}], TraditionalForm]](../HTMLFiles/index_158.gif){kind=small}
in the direction of 
.
Here's the same picture of 
with unit vectors 
and 
![FormBox[Cell[], TraditionalForm]](../HTMLFiles/index_163.gif){kind=small}
in the directions of 
and 
.
The matrix 
sends
to the direction of 
and
to the direction of 
.
Since 
you see that 
stretches vectors in the direction of 
by a factor of 
.
Since 
you see that 
stretches vectors in the direction of 
by a factor of 
.
The numbers 
and 
are called the singular values of the matrix 
.
Here's the picture of 
for another 
matrix 
.
The graphic shows that 
sends 
and 
to the directions of 
and 
.
Move 
to 
to compute 
.
Move 
to 
to compute 
.
(Q11) The singular values of 
are __ and __.
You can use the singular values of 
to compute 
.
will stretch 
by 3 and send it to 
.
will stretch 
by 
and send it to 
.
So 
.
(Q12) Move 
to 
to verify 
.
Here's the picture of 
for another 
matrix 
.
The matrix 
sends 
and 
to the directions of 
and 
.
This time the vector 
is stuck at 
.
(Q13) The graphic shows
.
How much did 
stretch 
? How much did 
stretch 
?
(Q14) The singular values of 
are __ and __.
(Q15) Use the singular values of 
to compute 
.
Copyright © 2007 Todd Will
Created by
Mathematica (April 15, 2007)