Least Squares Solutions by the numbers

Here's the picture of xA x for the 2  2 matrix A = (v    v ) (s      ) (u ) = ( 3  4  ) (0   0) ( 5   12  )   ...           --- --                                           5 5                               13 13.  

You see that  s_1 = 0 and s_2 = 2.  
Since s_1 = 0 you know that every vector x gets sent to a multiple of v_2.
Your job is to solve A x = y = (2)             3.

Step 1:  Find the v_i coordinates for y .  
From the plot it appears that y≈ -1v_1 + 3v_2.
Compute the exact coordinates:
    y = (v_1  y) v_1 + (v_2  y) v_2 = (6/5 - 12/5) v_1 + (8/5 + 9/5) v_2 = -6/5 v_1 + 17/5v_2.
Since s_1 = 0, throw away the v_1 portion of y to get Overscript[y,^] = 17/5v_2.
    Continue solving A x = Overscript[y,^].

Step 2:  Divide 17/5 by the singular value 2 to get  17/10.

Step 3:  Use this number as the u_2 coordinate for x to get x = 17/10u_2.

To get a numerical solution multiply out:  x = 17/10u_2 = 17/10 (-12) = (  102 )                       ---              - ...             17                       --               --                       13               26.
So one solution to A x = Overscript[y,^] is x = (  102 )               ----                65                17               --               26.  
Since s_1 = 0, all solution to A x = Overscript[y,^]  are given by
    x = (  102 ) + s (5 ) = (  102 ) + t (5 )                ----  ...        --                  --              --               26                  13              26.

Final answer:  There are NO solutions to A x = (2)          3.  But the least squares solutions are x = (  102 ) + t (5 )               ----                65                 12                17               --               26.

Here's the picture of xA x for the 2  2 matrix A = (v    v ) (s      ) (u ) = ( 7   24  ) (0   0) ( 3  4  )   ...             -- -                                           25 25                               5 5.  

Your job is to solve A x = y = (4)             2.

Step 1:  Find the v_i coordinates for y .  
From the plot it appears that y≈ -1v_1 + 4.5v_2.
Since {v_1, v_2} = {(7  ), (24)}                --     --                25     25                 -24    7                ---    --                25     25 is an orthonormal basis, you can compute the exact coordinates by using dot products.

(Q37)  y = ___ v_1 + ___ v_2

Since s_1 = 0, the projection of y on the column space of A is

(Q38)  Overscript[y,^] = ___ v_1 + ___ v_2

(Q39) How many solutions will A x = y have?  ___
How many solutions will A x = Overscript[y,^] have?   ___

Step 2:  Divide by the singular value.

Step 3:  All least squares solution to A x = (4)         2 are given by

(Q40)  x = ___ u_1 + ___ u_2

(Q41) Multiply out to get a numerical answer.  There are NO solutions to A x = (2)          3.  But the least squares solutions are x =


Up

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