Singular Values

Here's the picture of xA x for another 2  2 matrix A.

The plot on the left shows a grid spanned by an orthonormal basis {u_1, u_2}.
The matrix A sends this grid to an orthogonal grid on the right spanned by {A u_1, A u_2}.  
The grid on the left has 1  1 squares, but the grid on the right has 3  2 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]in the direction of {A u_1, A u_2}.

Here's the same picture of xA x with unit vectors v_1and v_2 FormBox[Cell[], TraditionalForm]in the directions of A u_1and A u_2

The matrix A sends
     u_1 to the direction of v_1 and
     u_2 to the direction of v_2.
Since A u_1 = 3v_1 you see that A stretches vectors in the direction of u_1 by a factor of 3.
Since A u_2 = 2v_2 you see that A stretches vectors in the direction of u_2 by a factor of 2.
The numbers 3 and 2 are called the singular values of the matrix A.

Here's the picture of xA x for another 2  2 matrix A.

The graphic shows that A sends u_1and u_2 to the directions of v_1and v_2.

Move x to u_1 to compute A u_1 = __ v_1.
Move x to u_2 to compute A u_2 = __ v_2.

(Q11) The singular values of A are __ and __.

You can use the singular values of A to compute A(u_1 + 2u_2).
    A will stretch u_1by 3 and send it to 3v_1.
    A will stretch 2u_2by 3/2 and send it to (3/2) 2v_2 = 3v_2.
So A(u_1 + 2u_2) = 3v_1 + (3/2) 2v_2 = 3v_1 + 3v_2.

(Q12) Move x to u_1 + 2u_2 to verify A(u_1 + 2u_2) = __ v_1 + __ v_2.

Here's the picture of xA x for another 2  2 matrix A.

The matrix A sends u_1and u_2 to the directions of v_1and v_2.
This time the vector x is stuck at x = u_1 + u_2.  

(Q13) The graphic shows
    A x = A u_1 + A u_2 = __ v_1 + __ v_2.   

How much did A stretch u_1?  How much did A stretch u_2?

(Q14) The singular values of A are __ and __.

(Q15) Use the singular values of A to compute A(3u_1 + 5u_2) = __ v_1 + __ v_2.


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Copyright 2007 Todd Will
Created by Mathematica  (April 15, 2007)