Three steps to solve A x = y

Here's the picture of xA x for the 2 × 2 matrix A = (v v ) (4 0) (u ) 1 2 1 0 3 u 2.

You recognize that the diagonal entries of (4 0) 0 3are the singular values.  You can also use the graphic to verify the fundamental equations A u_1 = 4v_1 and A u_2 = 3v_2.

Your job is to solve A x = y.

Three steps to solving A x = y .

Step 1:  Find the v_i coordinates for y .  
    From the plot you see y = 2v_1 + (-3) v_2.
Step 2:  Divide the 2 and -3 by the singular values 4 and 3 to get  2/4, -3/3.
Step 3:  Use these numbers as u_i coordinates for x to get     
    x = 1/2u_1 + (-1) u_2.

Use the graphic to verify that x = 1/2u_1 - u_2 is a solution to A x = y

Here's the picture of xA x for the 2 × 2 matrix A = (v v ) (s ) (u ) 1 2 1 0 1 s u 0 2 2.

This time the vector x doesn't move, but the picture has all the info you need to solve A x = y.

(Q21) The graphic shows that the singular values are s_1 = __ and s_2 = __.

(Q22) Step 1: Find the v_i coordinates for y:   y = __  v_1 + __  v_2.

Step 2:  Divide the v_i coords by the singular values.

(Q23) Step 3:  Use these numbers as u_i coordinates.
The solution to A x = y is x = __ u_1 + __ u_2.

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Copyright © 2007 Todd Will
Created by Mathematica  (April 14, 2007)