This page requires a very high tolerance for mathematics.
This page is the actual derivations used to compute the examples used in the Least Squares page.
Some examples may still be being worked on.
*** Notation warning *** This example uses A' to mean A inverse.
Least squares example one: Multiplying through by the inverse
Problem:
[ W0+W1 -W1 ] [ P1 ] = [ W0 -W1 ] [ SO ]
[ -W1 +W1 ] [ P2 ] [ 0 +W1 ] [ S1 ]
Setup for inverse:
[ W0+W1 -W1 | 1 0 ]
[ -W1 +W1 | 0 1 ]
Row 2 added to row 1
[ W0 0 | 1 1 ]
[ -W1 +W1 | 0 1 ]
Row 1 scaled by W0'
[ 1 0 | W0' W0' ]
[ -W1 +W1 | 0 1 ]
Row 1 premultiplied by W1 added to row 2
[ 1 0 | W0' W0' ]
[ 0 +W1 | W1*W0' 1+W1*W0' ]
Row 2 scaled by premultiplication by W1'
[ 1 0 | W0' W0' ]
[ 0 1 | W0' W1'+W0' ]
Row to cleaned up
[ 1 0 | W0' W0' ]
[ 0 1 | W0' W0'+W1' ]
And now premultiplying both sides by the inverse we get.
[ P1 ] = [ W0' W0' ] [ W0 -W1 ] [ SO ]
[ P2 ] = [ W0' W0'+W1' ] [ 0 +W1 ] [ S1 ]
=
points (Shot view) Shots
[ P1 ] = [ W0' W0' ] [ W0 -W1 ] [ SO ]
[ P2 ] [ W0' W0'+W1' ] [ 0 +W1 ] [ S1 ]
=
[ P1 ] = [ W0'W0 -W0'W1+W0'W1 ] [ S0 ]
[ P2 ] [ W0'W0 -W0'W1+(W0'+W1')W1 ] [ S1 ]
W0'W0 = 1
-W0'W1+W0'W1 is 0
-W0'W1+(W0'+W1')W1 is -W0'W1 + W0'W1 + W1'W1
which is W1'W1
which is 1
=
Shot view Point view
[ P1 ] = [ 1 0 ] [ S0 ] = [ S0 ]
[ P2 ] [ 1 1 ] [ S1 ] [ S0+S1 ]
*** Notation warning *** This example uses A' to mean A inverse.
Least squares example two, multiplying through by the inverse.
Problem:
[ W0+W1+W2 -W1-W2 ] [ P1 ] = [ W0 -W1 -W2 ] [ SO ]
[ -W1-W2 W1+W2 ] [ P2 ] [ 0 W1 W2 ] [ S1 ]
[ S2 ]
Setup to find inverse:
[ W0+W1+W2 -W1-W2 | 1 0 ]
[ -W1-W2 W1+W2 | 0 1 ]
Row 2 added to row 1
[ W0 0 | 1 1 ]
[ -W1-W2 W1+W2 | 0 1 ]
Row 1 scaled by W0'
[ 1 0 | W0' W0' ]
[ -W1-W2 W1+W2 | 0 1 ]
Row 1 premultiplied by (W1+W2)' added to row 2
[ 1 0 | W0' W0' ]
[ 0 W1+W2 | W0' 1+(W1+W2)'W0' ]
Scale row 2 premultiplying by (W1+W2)'
[ 1 0 | W0' W0' ]
[ 0 1 | W0' (W1+W2)'+W0' ]
Cleaning up row 2 slightly.
[ 1 0 | W0' W0' ]
[ 0 1 | W0' W0'+(W1+W2)' ]
So multiplying both sides by the inverse we have =
[ P1 ] = [ W0' W0' ] [ W0 -W1 -W2 ] [ SO ]
[ P2 ] [ W0' W0'+ (W1+W2)' ] [ 0 W1 W2 ] [ S1 ]
[ S3 ]
Everything up to the shots on the right hand side give the
view in terms of shots...
[ P1 ] = [ W0'W0 -W0'W1+W0'W1 -W0'W2+W0'W2 ] [ SO ]
[ P2 ] [ W0'W0 -W0'W1+(W0'+(W1+W2)')W1 -W0'W2+(W0'+(W1+W2)')W2 ] [ S1 ]
[ S3 ]
Grim, No?
W0'W0 is 1
-W0'W1+W0'W1 is 0
-W0'W2+W0'W2 is 0
-W0'W1+(W0'+(W1+W2)'+W0')W1 is -W0'W1 +W0'W1 + (W1+W2)'W1
Which is (W1+W2)'W1
-W0'W2+(W0'+(W1+W2)')W2 is -W0'W2 + W0W2 + (W1+W2)'W2
Which is (W1+W2)'W2
Giving the view as shots:
[ P1 ] = [ 1 0 0 ] [ SO ]
[ P2 ] [ 1 (W1+W2)'W1 (W1+W2)'W2 ] [ S1 ]
[ S3 ]
And the view as points:
[ P1 ] = [ S0 ]
[ P2 ] [ S0 + (W1+W2)'W1S1 + (W1+W2)'W2S2 ]
*** Notation warning *** This example users A' to mean A inverse.
Least squares example three, multiplying both sides by the inverse.
Problem:
[ WO+W1 -W1 0 0 ] [ P1 ] [ W0 -W1 0 0 0 ] [ SO ]
[ -W1 +W1+W2+W4 -W2 -W4 0 ] [ P2 ] = [ 0 W1 -W2 0 W4 ] [ S1 ]
[ 0 -W2-W4 +W2+W3+W4 -W3 ] [ P3 ] [ 0 0 W2 -W3 -W4 ] [ S2 ]
[ 0 0 -W3 +W3 ] [ P4 ] [ 0 0 0 W3 0 ] [ S3 ]
[ S4 ]
Setup to find the inverse:
[ WO+W1 -W1 0 0 | 1 0 0 0 ]
[ -W1 +W1+W2+W4 -W2 -W4 0 | 0 1 0 0 ]
[ 0 -W2-W4 +W2+W3+W4 -W3 | 0 0 1 0 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 4 added to row 3
[ WO+W1 -W1 0 0 | 1 0 0 0 ]
[ -W1 +W1+W2+W4 -W2 -W4 0 | 0 1 0 0 ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 3 added to row 2
[ WO+W1 -W1 0 0 | 1 0 0 0 ]
[ -W1 +W1 0 0 | 0 1 1 1 ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 2 added to row 1
[ WO 0 0 0 | 1 1 1 1 ]
[ -W1 +W1 0 0 | 0 1 1 1 ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 1 premultiplied by W0'
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ -W1 +W1 0 0 | 0 1 1 1 ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 1 premultiplied by W1 added to row 2
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 +W1 0 0 | W1W0' W1W0'+1 W1W0'+1 W1W0'+1 ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 2 premultiplied by W1'
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 1 0 0 | W0' W0'+W1' W0'+W1' W0'+W1' ]
[ 0 -W2-W4 +W2 +W4 0 | 0 0 1 1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 2 premultiplied by (W2+W4) added to row 3
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 1 0 0 | W0' W0'+W1' W0'+W1' W0'+W1' ]
[ 0 0 +W2+W4 0 | (W2+W4)W0' (W2+W4)(W0'+W1') (W2+W4)(W0'+W1')+1 (W2+W4)(W0'+W1')+1 ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 3 premultiplied by (W2+W4)'
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 1 0 0 | W0' W0'+W1' W0'+W1' W0'+W1' ]
[ 0 0 1 0 | W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)' ]
[ 0 0 -W3 +W3 | 0 0 0 1 ]
Row 3 premultiplied by W3 added to row 4
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 1 0 0 | W0' W0'+W1' W0'+W1' W0'+W1' ]
[ 0 0 1 0 | W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)' ]
[ 0 0 0 W3 | W3W0' W3(W0'+W1') W3(W0'+W1'+(W2+W4)') W3(W0'+W1'+(W2+W4)')+1 ]
premultiply row 4 by W3'
[ 1 0 0 0 | W0' W0' W0' W0' ]
[ 0 1 0 0 | W0' W0'+W1' W0'+W1' W0'+W1' ]
[ 0 0 1 0 | W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)' ]
[ 0 0 0 1 | W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)'+W3' ]
Ok, we have the inverse, multiplying both sides through by it gives:
[ P1 ] [ W0' W0' W0' W0' ] [ W0 -W1 0 0 0 ] [ SO ]
[ P2 ] = [ W0' W0'+W1' W0'+W1' W0'+W1' ] [ 0 W1 -W2 0 W4 ] [ S1 ]
[ P3 ] [ W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)' ] [ 0 0 W2 -W3 -W4 ] [ S2 ]
[ P4 ] [ W0' W0'+W1' W0'+W1'+(W2+W4)' W0'+W1'+(W2+W4)'+W3' ] [ 0 0 0 W3 0 ] [ S3 ]
[ S4 ]
The first two matrices on the right hand side multiply out to:
[ 1 0 0 0 0 ]
[ 1 1 0 0 0 ]
[ 1 1 (W2+W4)'W2 0 -(W2+W4)'W4 ]
[ 1 1 (W2+W4)'W2 1 -(W2+W4)'W4 ]
So we have
(Shots as the point of view) (Points as the point of view)
[ P1 ] [ 1 0 0 0 0 ] [ SO ] [ S0 ]
[ P2 ] = [ 1 1 0 0 0 ] [ S1 ] = [ S0 + S1 ]
[ P3 ] [ 1 1 (W2+W4)'W2 0 -(W2+W4)'W4 ] [ S2 ] [ S0 + S1 + (W2+W4)'W2S2 - (W2+W4)'W4S4 ]
[ P4 ] [ 1 1 (W2+W4)'W2 1 -(W2+W4)'W4 ] [ S3 ] [ S0 + S1 + (W2+W4)'W2S2 + S3 - (W2+W4)'W4S4 ]
[*** Under construction ***]
*** Notation Warning *** This example uses A' to mean inverse(A)
Least squares example four, multiplying both sides by the inverse.
Problem:
[ W0+W1+W3 -W1 -W3 ] [ P1 ] = [ W0 -W1 0 W3 ] [ S0 ]
[ -W1 W1+W2 -W2 ] [ P2 ] [ 0 W1 -W2 0 ] [ S1 ]
[ -W3 -W2 +W2+W3 ] [ P3 ] [ 0 0 W2 -W3 ] [ S2 ]
[ S3 ]
Setting up for the inverse:
[ W0+W1+W3 -W1 -W3 | 1 0 0 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Add row 3 to row 1
[ W0+W1 -W1-W2 +W2 | 1 0 1 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Add row 2 to row 1
[ W0 0 0 | 1 1 1 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
-------- Alternate derivation 1
Row 1 premultiplied by W1W0' added to row 2.
[ W0 0 0 | 1 1 1 ]
[ 0 W1+W2 -W2 | W1W0' 1+W1W0' W1W0' ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W1W0' added to row 2.
[ W0 0 0 | 1 1 1 ]
[ 0 W1+W2 -W2 | W1W0' 1+W1W0' W1W0' ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W3W0' added to row 3
[ W0 0 0 | 1 1 1 ]
[ 0 W1+W2 -W2 | W1W0' 1+W1W0' W1W0' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' 1+W3W0' ]
Row 3 added to row 2
[ W0 0 0 | 1 1 1 ]
[ 0 W1 W3 | W3W0'+W1W0' W3W0'+1+W1W0' W3W0'+W1W0' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' 1+W3W0' ]
Row 2 premultiplied by W2W1' added to row 3
[ W0 0 0 | 1 1 1 ]
[ 0 W1 W3 | W3W0'+W1W0' W3W0'+ 1+W1W0' W3W0'+W1W0' ]
[ 0 0 +W2+W3+W2W1'W3 | W3W0'+W2W1'W3W0'+W2W0' W3W0'+W2W1'W3W0'+W2W1'+W2W0' 1+W3W0'+W2W1'W3W0'+W2W0' ]
+W2+W3+W2W1'W3 = ( W2W3'+ 1 + W2W1') W3
= W2 (W3'+ W2'+ W1') W3
= W2(W1'+W2'+W3')W3
W3W0'+W2W1'W3W0'+W2W0' = (W3+W2W1'W3+W2)W0'
= (W2+W2W1'W3+W3)W0'
= W2(1+ W1'W3+W2'W3)W0'
= W2(W3'+W1'+W2')W3W0'
= W2(W1'+W2'+W3')W3W0'
W3W0'+W2W1'W3W0'+W2W1'+W2W0' = (W3+W2W1'W3+W2W1'W0+W2)W0'
= (W2+W2W1'W3+W2W1'W0+W3)W0'
= W2(1+W1'W3+W2W1'W0+W2'W3)W0'
= W1(W3'+W1'+W2W1'W0W3'+W2')W3W0'
or
W3W0'+W2W1'W3W0'+W2W1'+W2W0' = (W3+W2W1'W3+W2)W0'+W2W1'
= (W2+W2W1'W3+W3)W0'+W2W1'
= W2(W2'+W1'+W3')W3W0'+W2W1'
= W2(W1'+W2'+W3')W3W0'+W2W1'
1+W3W0'+W2W1'W3W0'+W2W0' = 1+(W3+W2W1'W3+W2)W0'
= 1+(W2+W2W1'W3+W3)W0'
= 1+W2(1+W1'W3+W2'W3)W0'
= 1+W2(W3'+W1'+W2')W3W0'
= 1+W2(W1'+W2'+W3')W3W0'
[ W0 0 0 | 1 1 1 ]
[ 0 W1 W3 | W3W0'+W1W0' W3W0'+1+W1W0' W3W0'+W1W0' ]
[ 0 0 W2(W1'+W2'+W3')W3 | W2(W1'+W2'+W3')W3W0' W2(W1'+W2'+W3')W3W0'+W2W1' W2(W1'+W2'+W3')W3W0'+1 ]
Scale row 3 premultiplying by (W2(W1'+W2'+W3')W3)' = W3'(W1'+W2'+W3')'W2'
[ W0 0 0 | 1 1 1 ]
[ 0 W1 W3 | W3W0'+W1W0' W3W0'+1+W1W0' W3W0'+W1W0' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
Row 3 , premultiplied by -W3 added to row 2
[ W0 0 0 | 1 1 1 ]
[ 0 W1 0 | W3W0'+W1W0'-W3W0' W3W0'+1+W1W0'-W3W0'-(W1'+W2'+W3')'W1' W3W0'+W1W0'-W3W0'-(W1'+W2'+W3')'W2' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
=
[ W0 0 0 | 1 1 1 ]
[ 0 W1 0 | W1W0' 1+W1W0'-(W1'+W2'+W3')'W1' W1W0'-(W1'+W2'+W3')'W2' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
Scaling row 2 by premultiplying by W1'
[ W0 0 0 | 1 1 1 ]
[ 0 1 0 | W0' W0'+W1'-W1'(W1'+W2'+W3')'W1' W0'-W1'(W1'+W2'+W3')'W2' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
Scaling row 1 by premultiplying by W0'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 0 | W0' W0'+W1'-W1'(W1'+W2'+W3')'W1' W0'-W1'(W1'+W2'+W3')'W2' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
W0'+W1'-W1'(W1'+W2'+W3')'W1' = W0'+(W2'+W3')(W1'+W2'+W3')W1'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 0 | W0' W0'+(W2'+W3')(W1'+W2'+W3')W1' W0'-W1'(W1'+W2'+W3')'W2' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'W2' ]
*** Note to myself
c(a+b+c)'a - a(a+b+c)'c
ca'(a'+(b+c)')'a - a(a'+(b+c)')'a'c
c(a'(a'+(b+c)')'a - c'a(a'+(b+c)')'a'c)
c(a'(a'+(b+c)')'ac' - c'a(a'+(b+c)')'a')c
abc-cba = a'b'c'-cba = (cba)'-cba = I - cbacba
*** End note.
-------- End alternate derivation 1
--------- Alternate derivation 2
Setting up for the inverse:
[ W0+W1+W3 -W1 -W3 | 1 0 0 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Add row 3 to row 1
[ W0+W1 -W1-W2 +W2 | 1 0 1 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Add row 2 to row 1
[ W0 0 0 | 1 1 1 ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Scale row 1 by premultiplying W0'
[ 1 0 0 | W0' W0' W0' ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W1 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1+W2 -W2 | W1W0' W1W0'+1 W1W0' ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W3 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1+W2 -W2 | W1W0' W1W0'+1 W1W0' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Row 3 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1 W3 | W1W0'+W3W0' W1W0'+W3W0'+1 W1W0'+W3W0'+1 ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
**** Derivation 3
Row 2 premultiplied by W1'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Row 1 premultiplied by W2 added to row three
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 0 W2W1'W3++W2+W3 | W2W0'+W2W1'W3W0'+W3W0' W2W0'+W2W1'W3W0'+W2W1'+W3W0' W2W0'+W2W1'W3W0'+W2W1'+W3W0'+1 ]
=
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 0 W1'W3+1+W2'W3 | W0'+W1'W3W0'+W2'W3W0' W0'+W1'W3W0'+W1'+W2'W3W0' W0'+W1'W3W0'+W1'+W2'W3W0'+W2' ]
=
****
Row 2 scaled by premultiplying by W1'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Row 2 premultiplied by W2 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+ W1'W3W0' W0' +W1'W3W0' +W1' W0' +W1'W3W0' +W1' ]
[ 0 0 +W2+W3-W2W1'W3 | W2W0'+W2W1'W3W0'+W3W0' W2W0'+W2W1'W3W0'+W2W1'+W3W0' W2W0'+W2W1'W3W0'+W2W1'+W3W0'+1 ]
+W2+W3+W2W1'W3 = W2(W1'+W2'+W3')W3
-W2W0'-W2W1'W3W0'+W3W0' = (W2+W2W1'W3+W3)W0'
= W2(W1'+W2'+W3')W3W0'
W2W0'+W2W1'W3W0'+W2W1'+W3W0' = (W2+W2W1'W3+W3)W0'+W2W1'
= W2(W1'+W2'+W3')W3W0+W2W1'
W2W0'+W2W1'W3W0'+W2W1'+W3W0'+1 = W2(W1'+W2'+W3')W3W0'+W2W1'+1
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0' +W1'W3W0' +W1' ]
[ 0 0 W2(W1'+W2'+W3')W3 | W2(W1'+W2'+W3')W3 W2(W1'+W2'+W3')W3W0+W2W1' W2(W1'+W2'+W3')W3W0'+W2W1'+1 ]
Scaling row 3 by premultiplying by (W2(W1'+W2'+W3')W3)' = W3'(W1'+W2'+W3')'W2'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W2'W2W1' W0'+W3'(W1'+W2'+W3')'W2'W2W1'+W3'(W1'+W2'+W3')'W2' ]
W0'+W3'(W1'+W2'+W3')'W2'W2W1' = W0'+W3'(W1'+W2'+W3')'W1'
W0'+W3'(W1'+W2'+W3')'W2'W2W1'+W3'(W1'+W2'+W3')'W2'
= W0'+W3'(W1'+W2'+W3')'W1'+W3'(W1'+W2'+W3')'W2'
= W0'+W3'(W1'+W2'+W3')'(W1'+W2')
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'W3W0'+W1' W0'+W1'W3W0'+W1' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1' W0'+W3'(W1'+W2'+W3')'(W1'+W2') ]
Row 3 premultiplied by -W3'W1 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0'-W1'W3W0' W0'+W1'W3W0'+W1'-W1'W3W0'-W1'W3W3'(W1'+W2'+W3')'W1 W0'+W1'W3W0'+W1'-W1'W3W0'-W1'W3W3'(W1'+W2'+W3')'(W1'+W2') ] ]
[ 0 0 1 | W0' W0' +W3'(W1'+W2'+W3')'W1' W0' +W3'(W1'+W2'+W3')'(W1'+W2') ]
W0'+W1'W3W0'-W1'W3W0' = W0'
W0'+W1'W3W0'+W1'-W1'W3W0'-W1'W3W3'(W1'+W2'+W3')'W1 = W0'+W1'-W1'(W1'+W2'+W3')'W1
W0'+W1'W3W0'+W1'-W1'W3W0'-W1'W3W3'(W1'+W2'+W3')'(W1'+W2') = W0'+W1'-W1'(W1'+W2'+W3')'(W1'+W2')
--------- End alternate derivation 2
Row 1 scaled premultiplying by W0'
[ 1 0 0 | W0' W0' W0' ]
[ -W1 W1+W2 -W2 | 0 1 0 ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W1 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1+W2 -W2 | W1W0' W1W0'+1 W1W0' ]
[ -W3 -W2 +W2+W3 | 0 0 1 ]
Row 1 premultiplied by W3 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1+W2 -W2 | W1W0' W1W0'+1 W1W0' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
*** Alt 5
pre-Scale 2 by (W1+W2)'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 -(W1+W2)'W2 | (W1+W2)'W1W0' (W1+W2)'W1W0'+1 (W1+W2)'W1W0' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Add W2 * row 2 to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 -(W1+W2)'W2 | (W1+W2)'W1W0' (W1+W2)'W1W0'+1 (W1+W2)'W1W0' ]
[ 0 0 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
***
Row 3 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 W1 W3 | W1W0'+W3W0' W1W0'+1+W3W0' W1W0'+W3W0'+1 ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Scale row 2 by premultiplying by W1' And canceling out W1'W1 terms.
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'+W1'W3W0' W0'+W1'W3W0'+W1' ]
[ 0 -W2 +W2+W3 | W3W0' W3W0' W3W0'+1 ]
Scale row 3 by premuliplying by W2'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'+W1'W3W0' W0'+W1'W3W0'+W1' ]
[ 0 -1 W2'(W2+W3) | W2'W3W0' W2'W3W0' W2'W3W0'+W2' ]
Rearrange row 2
W0'+W1'W3W0'+W1' = W0'+W1'+W1'W3W0'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'+W1'W3W0' W0'+W1'+W1'W3W0' ]
[ 0 -1 W2'(W2+W3) | W2'W3W0' W2'W3W0' W2'W3W0'+W2' ]
Add row 2 to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | W0'+W1'W3W0' W0'+W1'+W1'W3W0' W0'+W1'+W1'W3W0' ]
[ 0 0 W2'(W2+W3)+W1'W3 | W2'W3W0'+W0'+W1'W3W0' W2'W3W0'+W0'+W1'+W1'W3W0' W2'W3W0'+W2'+W0'+W1'+W1'W3W0' ]
Cleanup row 2
W0'+W1'W3W0' = (1+W1'W3)W0'
= (W3'+W1')W3W0'
W0'+W1'+W1'W3W0' = (1+W1'W0+W1'W3)W0'
= W1'(W1+W0+W3)W0'
Cleanup row 3
W2'(W2+W3)+W1'W3 = (W2'+W3')W3 + W1'W3 [ A'(A+B) = (A'+B')B ]
= (W2'+W3'+W1')W3 [ right distribution ]
= (W1'+W2'+W3')W3 [ Addition commutes. ]
W2'W3W0'+W0'+W1'W3W0' = (W2'W3 + 1 + W1'W3)W0' [ Right distribution ]
= ((W2'+W3'+W1')W3W0' [ Right distribution ]
= (W1'+W2'+W3')W3W0' [ Association ]
W2'W3W0'+W0'+W1'+W1'W3W0' = (W2'W3 + 1 + W1'W0 + W1'W3)W0' [ Right distribution ]
= (W2'W3 + W1'W3 + W1'W0 + 1)W0' [ Addition commutes ]
= (W2'+W1'+W1'W0W3'+W3')W3W0' [ Right distribution ]
= (W1'+W2'+W3'+W1'W0W3')W3W0' [ Additon commutes ]
= (W1'+W2'+W3')W3W0' + W1'W0W3'W3W0' [ Right distribution ]
= (W1'+W2'+W3')W3W0' + W1'W0' [ Canceling inverses ]
W2'W3W0'+W2'+W0'+W1'+W1'W3W0' = W2'W3W0'+ W0'+W1'+W1'W3W0' + W2' [ Addition commutes ]
= (W1'+W2'+W3')W3W0' + W1'W0' + W2' [ Previous derivation ]
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | (W3'+W1')W3W0' W1'(W1+W0+W3)W0' W1'(W1+W0+W3)W0' ]
[ 0 0 (W1'+W2'+W3')W3 | (W1'+W2'+W3')W3W0' (W1'+W2'+W3')W3W0'+W1'W0' (W1'+W2'+W3')W3W0'+W1'W0'+W2' ]
Scaling row three by premultiplying by (W1'+W2'+W3')'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | (W3'+W1')W3W0' W1'(W1+W0+W3)W0' W1'(W1+W0+W3)W0' ]
[ 0 0 W3 | W3W0' W3W0'+(W1'+W2'+W3')'W1'W0' W3W0'+(W1'+W2'+W3')'(W1'W0'+W2') ]
Scaling row three by premultiplying by W3'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 W1'W3 | (W3'+W1')W3W0' W1'(W1+W0+W3)W0' W1'(W1+W0+W3)W0' ]
[ 0 0 1 | W0' W0'+W3'(W1'+W2'+W3')'W1'W0' W0'+W3'(W1'+W2'+W3')'(W1'W0'+W2') ]
*** Notes to myself
What can we simplify?
\(\A+\B)
= A - A \(A+B) A
= B - B \(A+B) B
\A \(\A+\B) = \(A+B) B
\(\A+\B) = A \(A+B) B
*** End notes.
Row three premultiplied by -W3'W1 added to row 2
=========================================
*Clearly wrong. The original is symmetric, so the inverse must be. *
The old file (lsexamples.html.old) came out symmetric, so I know it is a
problem in converting the notation over...
***** WORKING ****
[ W0+W1+W3 -W1 -W3 ] [ P1 ] = [ W0 -W1 0 W3 ] [ S0 ]
[ -W1 W1+W2 -W2 ] [ P2 ] [ 0 W1 -W2 0 ] [ S1 ]
[ -W3 -W2 +W2+W3 ] [ P3 ] [ 0 0 W2 -W3 ] [ S2 ]
[*** Under construction ***]
Least squares example five, by elementary row transformations.
Problem:
[ W0+W1+W2+W3 -W1-W2 -W3 ] [ P1 ] [ WO -W1 -W2 -W3 0 ] [ SO ]
[ -W1-W2 +W1+W2+W4 -W4 ] [ P2 ] = [ 0 W1 W2 0 W4 ] [ S1 ]
[ -W3 -W4 +W3+W4 ] [ P3 ] [ 0 0 0 W3 -W4 ] [ S2 ]
[ S3 ]
[ S4 ]
Setup for inverse:
[ W0+W1+W2+W3 -W1-W2 -W3 | 1 0 0 ]
[ -W1-W2 +W1+W2+W4 -W4 | 0 1 0 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
-------- ALT 1
Add row 3 to row 1
[ W0+W1+W2 -W1-W2-W4 +W4 | 1 0 1 ]
[ -W1-W2 +W1+W2+W4 -W4 | 0 1 0 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Add row 2 to row 1
[ W0 0 0 | 1 1 1 ]
[ -W1-W2 +W1+W2+W4 -W4 | 0 1 0 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Scale row 1 by premultiplying by W0'
[ 1 0 0 | W0' W0' W0' ]
[ -W1-W2 +W1+W2+W4 -W4 | 0 1 0 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 1 premultiplied by W1+W2 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2+W4 -W4 | (W1+W2)W0' 1+(W1+W2)W0' (W1+W2)W0' ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 1 premultiplied by W3 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2+W4 -W4 | (W1+W2)W0' 1+(W1+W2)W0' (W1+W2)W0' ]
[ 0 -W4 +W3+W4 | W3W0' W3W0' W3W0'+1 ]
Row 3 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W3W0'+(W1+W2)W0' W3W0'+1+(W1+W2)W0' W3W0'+1+(W1+W2)W0' ]
[ 0 -W4 +W3+W4 | W3W0' W3W0' W3W0'+1 ]
-------- ALT 3
Row 2 premultiplied by W4(W1+W2)' added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W3W0'+(W1+W2)W0' W3W0'+1+(W1+W2)W0' W3W0'+1+(W1+W2)W0' ]
[ 0 0 W4(W1+W2)'W3+W3+W4 | W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'(W1+W2)W0' W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4(W1+W2)'(W1+W2)W0' W3W0'+1+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4(W1+W2)'(W1+W2)W0' ]
W4(W1+W2)'W3+W3+W4 = W4(W1+W2)'W3+W4(W3'+W4')W3
= W4((W1+W2)'+W3'+W4')W3
W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'(W1+W2)W0' = W3W0'+W4(W1+W2)'W3W0'+W4W0'
= (W3+W4(W1+W2)'W3+W4)W0'
= W4((W1+W2)'+W3'+W4')W3W0'
W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4(W1+W2)'(W1+W2)W0' = W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0'
= (W3+W4(W1+W2)'W3+W4)W0'+W4(W1+W2)'
= W4((W1+W2)'+W3'+W4')W3W0'+W4(W1+W2)'
W3W0'+1+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4(W1+W2)'(W1+W2)W0' = W3W0'+1+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0'
= (W3+W4(W1+W2)'W3+W4)W0'+W4(W1+W2)'+1
= W4((W1+W2)'+W3'+W4')W3W0'+W4(W1+W2)'+1
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W3W0'+(W1+W2)W0' W3W0'+1+(W1+W2)W0' W3W0'+1+(W1+W2)W0' ]
[ 0 0 W4((W1+W2)'+W3'+W4')W3 | W4((W1+W2)'+W3'+W4')W3W0' W4((W1+W2)'+W3'+W4')W3W0'+W4(W1+W2)' W4((W1+W2)'+W3'+W4')W3W0'+W4(W1+W2)'+1 ]
Premultiply row 3 by ( W4((W1+W2)'+W3'+W4')W3 )' = W3'(W1+W2)'+W3'+W4')'W4'
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W3W0'+(W1+W2)W0' W3W0'+1+(W1+W2)W0' W3W0'+1+(W1+W2)W0' ]
[ 0 0 1 | W0' W0'+W3'(W1+W2)'+W3'+W4')'W4'W4(W1+W2)' W0'+W3'((W1+W2)'+W3'+W4')'W4'W4(W1+W2)'+W3'(W1+W2)'+W3'+W4')'W4' ]
W0'+W3'(W1+W2)'+W3'+W4')'W4'W4(W1+W2)' = W0'+W3'(W1+W2)'+W3'+W4')'(W1+W2)'
W0'+W3'(W1+W2)'+W3'+W4')'W4'W4(W1+W2)'+W3'(W1+W2)'+W3'+W4')'W4'
= W0'+W3'(W1+W2)'+W3'+W4')'(W1+W2)'+W3'(W1+W2)'+W3'+W4')'W4'
= W0'+W3'(W1+W2)'+W3'+W4')'((W1+W2)'+W4')
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 W3 | W3W0'+(W1+W2)W0' W3W0'+1+(W1+W2)W0' W3W0'+1+(W1+W2)W0' ]
[ 0 0 1 | W0' W0'+W3'(W1+W2)'+W3'+W4')'(W1+W2)' W0'+W3'(W1+W2)'+W3'+W4')'((W1+W2)'+W4') ]
Row 3 premultiplied by -W3
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 0 | W3W0'+(W1+W2)W0'-W3W0' W3W0'+1+(W1+W2)W0'-W3W0'-W3'(W1+W2)'+W3'+W4')'(W1+W2)' W3W0'+1+(W1+W2)W0' ]
[ 0 0 1 | W0' W0'+W3'(W1+W2)'+W3'+W4')'(W1+W2)' W0'+W3'(W1+W2)'+W3'+W4')'((W1+W2)'+W4') ]
--------- End alt 3
Row 2 scaled by premultiplying by (W1+W2)'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 (W1+W2)'W3 | (W1+W2)'W3W0'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' ]
[ 0 -W4 +W3+W4 | W3W0' W3W0' W3W0'+1 ]
Row 2 premultiplied by W4 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 (W1+W2)'W3 | (W1+W2)'W3W0'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' ]
[ 0 0 +W3+W4+W4(W1+W2)'W3 | W3W0'+W4(W1+W2)'W3W0'+W4W0' W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0' W3W0'+1+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0' ]
W3+W4+W4(W1+W2)'W3 =
= W4+W3+W4(W1+W2)'W3
= W4(W4'+W3')W3 + W4(W1+W2)'W3
= W4(W4'+W3'+W1+W2)W3
W3W0'+W4(W1+W2)'W3W0'+W4W0' = (W3+W4(W1+W2)'W3+W4)W0'
= W4(W4'+W3'+W1+W2)W3W0'
W3W0'+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0' = (W3+W4(W1+W2)'W3+W4)W0'+W4(W1+W2)'
= W4(W4'+W3'+W1+W2)W3W0'+W4(W1+W2)'
W3W0'+1+W4(W1+W2)'W3W0'+W4(W1+W2)'+W4W0' = (W3+W4(W1+W2)'W3+W4)W0'+W4(W1+W2)'+1
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 (W1+W2)'W3 | (W1+W2)'W3W0'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' ]
[ 0 0 W4(W4'+W3'+W1+W2)W3 | W4(W4'+W3'+W1+W2)W3W0' W4(W4'+W3'+W1+W2)W3W0'+W4(W1+W2)' (W3+W4(W1+W2)'W3+W4)W0'+W4(W1+W2)'+1 ]
Premultiply row 3 by ( W4(W4'+W3'+W1+W2)W3 )' = W3'(W4'+W3'+W1+W2)'W4'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 (W1+W2)'W3 | (W1+W2)'W3W0'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' ]
[ 0 0 1 | W0' W0'+W3'(W4'+W3'+W1+W2)'W4'W4(W1+W2)' W0'+W3'(W4'+W3'+W1+W2)'W4'W4(W1+W2)'+W3'(W4'+W3'+W1+W2)'W4' ]
W0'+W3'(W4'+W3'+W1+W2)'W4'W4(W1+W2)' = W0'+W3'(W4'+W3'+W1+W2)'W4'W4(W1+W2)'
= W0'+W3'(W4'+W3'+W1+W2)'(W1+W2)'
W0'+W3'(W4'+W3'+W1+W2)'W4'W4(W1+W2)'+W3'(W4'+W3'+W1+W2)'W4'
= W0'+W3'(W4'+W3'+W1+W2)'(W1+W2)'+W3'(W4'+W3'+W1+W2)'W4'
= W0'+W3'(W4'+W3'+W1+W2)'((W1+W2)'+W4')
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 (W1+W2)'W3 | (W1+W2)'W3W0'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' (W1+W2)'W3W0'+(W1+W2)'+W0' ]
[ 0 0 1 | W0' W0'+W3'(W4'+W3'+W1+W2)'(W1+W2)' W0'+W3'(W4'+W3'+W1+W2)'((W1+W2)'+W4') ]
Add
-------- END ALT 1
Row 3 added to row 2
[ W0+W1+W2+W3 -W1-W2 -W3 | 1 0 0 ]
[ -W1-W2-W3 +W1+W2 +W3 | 0 1 1 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 2 added to row 1
[ W0 0 0 | 1 1 1 ]
[ -W1-W2-W3 +W1+W2 +W3 | 0 1 1 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 1 pre or post multiplied by W0'
[ 1 0 0 | W0' W0' W0' ]
[ -W1-W2-W3 +W1+W2 +W3 | 0 1 1 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 1 * (W1+W2+W3) added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | 0 W0'(W1+W2+W3)+1 W0'(W1+W2+W3)+1 ]
[ -W3 -W4 +W3+W4 | 0 0 1 ]
Row 1 * W3 added to row 3
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W0'(W1+W2+W3) W0'(W1+W2+W3)+1 W0'(W1+W2+W3)+1 ]
[ 0 -W4 +W3+W4 | W0'W3 W0'W3 W0'W3+1 ]
Partial distribution of row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 +W1+W2 +W3 | W0'(W1+W2)+W0'W3 W0'(W1+W2)+W0'W3+1 W0'(W1+W2)+W0'W3+1 ]
[ 0 -W4 +W3+W4 | W0'W3 W0'W3 W0'W3+1 ]
Row 2 * (W1+W2)'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 +W3(W1+W2)' | W0'+W0'W3(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' ]
[ 0 -W4 +W3+W4 | W0'W3 W0'W3 W0'W3+1 ]
Row 3 * W4
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 +W3(W1+W2)' | W0'+W0'W3(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' ]
[ 0 -1 +W3W4'+1 | W0'W3W4' W0'W3W4' W0'W3W4'+W4 ]
Row 3 added to row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 +W3(W1+W2)' | W0'+W0'W3(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' ]
[ 0 0 +W3W4'+1+W3(W1+W2)' | W0'W3W4'+W0'+W0'W3(W1+W2)' W0'W3W4'+W0'+W0'W3(W1+W2)'+(W1+W2)' W0'W3W4'+W4+W0'+W0'W3(W1+W2)'+(W1+W2)' ]
Cleanup row 3
+W3W4'+1+W3(W1+W2)' = W3(W4'+(W1+W2)')+1
W0'W3W4'+W0'+W0'W3(W1+W2)' = W0'(W3W4'+1+W3(W1+W2)')
= W0'(W3(W4'+(W1+W2)')+1)
W0'W3W4'+W0'+W0'W3(W1+W2)'+(W1+W2)' = W0'(W3(W4'+(W1+W2)')+1) + (W1+W2)'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 +W3(W1+W2)' | W0'+W0'W3(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' ]
[ 0 0 W3(W4'+(W1+W2)')+1 | W0'(W3(W4'+(W1+W2)')+1) W0'(W3(W4'+(W1+W2)')+1)+(W1+W2)' W0'(W3(W4'+(W1+W2)')+1)+(W1+W2)' ]
Row 3 multiplied by (W3(W4'+(W1+W2)')+1)' abbreviated as u'
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 +W3(W1+W2)' | W0'+W0'W3(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' W0'+W0'W3(W1+W2)'+(W1+W2)' ]
[ 0 0 1 | W0' W0'+(W1+W2)'u' W0'+(W1+W2)'u' ]
Row 3 * +W3(W1+W2)' subtracted from row 2
[ 1 0 0 | W0' W0' W0' ]
[ 0 1 0 | W0' W0'+(W1+W2)'-(W1+W2)'u' W0'+(W1+W2)'-(W1+W2)'u' ]
[ 0 0 1 | W0' W0'+(W1+W2)'u' W0'+(W1+W2)'u' ]
*Clearly wrong... the original is symmetric, so the inverse has to be symmetric.*
*** REWORK ***
[ W0+W1+W2+W3 -W1-W2 -W3 ] [ P1 ] [ WO -W1 -W2 -W3 0 ] [ SO ]
[ -W1-W2 +W1+W2+W4 -W4 ] [ P2 ] = [ 0 W1 W2 0 W4 ] [ S1 ]
[ -W3 -W4 +W3+W4 ] [ P3 ] [ 0 0 0 W3 -W4 ] [ S2 ]
[ S3 ]
[ S4 ]
This page is http://www.cc.utah.edu/~nahaj/cave/survey/intro/lsexamples.html © Copyright 2000 by John Halleck, All Rights Reserved. This snapshot was last modified on January 24th, 2001