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>> theta = linspace(-pi,pi,101);
>> u = [ cos(theta); sin(theta) ];
>> u = [ cos(theta)
         sin(theta) ;
         ];
>> size( u )

ans =

     2   101

>> u(:,1)

ans =

   -1.0000
   -0.0000

>> format long
>> u(:,1)

ans =

  -1.00000000000000
  -0.00000000000000

>> format long exponential
>> u(:,1)

ans =

  -1.00000000000000
  -0.00000000000000

>> format short
>> A = [ 2, 1
        -1, 2 ];
>> y = A*u;
>> y(:,1)

ans =

   -2.0000
    1.0000

>> A*u(:,1)

ans =

   -2.0000
    1.0000

>> plot( y )
>> plot( y' )
>> plot( y(1,:), y(2,:) )
>> axis( 'square' )
>> B = [ 5, 4
         6, 5 ]

B =

     5     4
     6     5

>> z = B*u;
>> plot( z(1,:), z(2,:) )
>> axis('square')
>> C = [ cos(1), sin(1)
         -sin(1), cos(1) ]

C =

    0.5403    0.8415
   -0.8415    0.5403

>> w = C*u;
>> plot( w(1,:), w(2,:) )
>> axis('square')
>> C'*C

ans =

     1     0
     0     1

>> [UA, SA, VA] = svd( A )

UA =

   -0.8944    0.4472
    0.4472    0.8944


SA =

    2.2361         0
         0    2.2361


VA =

    -1     0
     0     1

>> 
>> A - UA*SA*VA'

ans =

  1.0e-015 *

    0.2220         0
         0         0

>> UA'*UA

ans =

    1.0000    0.0000
    0.0000    1.0000

>> VA*VA'

ans =

     1     0
     0     1

>> SA

SA =

    2.2361         0
         0    2.2361

>> [UB, SB, VB] = svd( B )

UB =

   -0.6340   -0.7733
   -0.7733    0.6340


SB =

   10.0990         0
         0    0.0990


VB =

   -0.7733   -0.6340
   -0.6340    0.7733

>> plot( z(1,:), z(2,:) )
>> axis('square')
>> 6^2+8^2

ans =

   100

>> axis( [-0.5,0.5, -0.5,0.5] )
>> grid
>> 0.05^2 + 0.05^2

ans =

    0.0050

>> sqrt(ans)

ans =

    0.0707

>> Sing = [ 1, 1
            1, 1 ]

Sing =

     1     1
     1     1

>> det( Sing )

ans =

     0

>> [  US, SS, VS ] = svd( Sing )

US =

   -0.7071   -0.7071
   -0.7071    0.7071


SS =

     2     0
     0     0


VS =

   -0.7071   -0.7071
   -0.7071    0.7071

>> ss = Sing*u;
>> plot( ss(:,1), ss(:,2) )
>> size(ss)

ans =

     2   101

>> plot( ss(:,1), ss(:,2) )
>> ss(1,:)

ans =

  Columns 1 through 7 

   -1.0000   -1.0608   -1.1174   -1.1697   -1.2173   -1.2601   -1.2979

  Columns 8 through 14 

   -1.3306   -1.3581   -1.3802   -1.3968   -1.4079   -1.4135   -1.4135

  Columns 15 through 21 

   -1.4079   -1.3968   -1.3802   -1.3581   -1.3306   -1.2979   -1.2601

  Columns 22 through 28 

   -1.2173   -1.1697   -1.1174   -1.0608   -1.0000   -0.9352   -0.8668

  Columns 29 through 35 

   -0.7949   -0.7199   -0.6420   -0.5617   -0.4790   -0.3946   -0.3085

  Columns 36 through 42 

   -0.2212   -0.1331   -0.0444    0.0444    0.1331    0.2212    0.3085

  Columns 43 through 49 

    0.3946    0.4790    0.5617    0.6420    0.7199    0.7949    0.8668

  Columns 50 through 56 

    0.9352    1.0000    1.0608    1.1174    1.1697    1.2173    1.2601

  Columns 57 through 63 

    1.2979    1.3306    1.3581    1.3802    1.3968    1.4079    1.4135

  Columns 64 through 70 

    1.4135    1.4079    1.3968    1.3802    1.3581    1.3306    1.2979

  Columns 71 through 77 

    1.2601    1.2173    1.1697    1.1174    1.0608    1.0000    0.9352

  Columns 78 through 84 

    0.8668    0.7949    0.7199    0.6420    0.5617    0.4790    0.3946

  Columns 85 through 91 

    0.3085    0.2212    0.1331    0.0444   -0.0444   -0.1331   -0.2212

  Columns 92 through 98 

   -0.3085   -0.3946   -0.4790   -0.5617   -0.6420   -0.7199   -0.7949

  Columns 99 through 101 

   -0.8668   -0.9352   -1.0000

>> ss(:,1)

ans =

   -1.0000
   -1.0000

>> ss(:,2)

ans =

   -1.0608
   -1.0608

>> ss(:,3)

ans =

   -1.1174
   -1.1174

>> plot(ss(:,1),ss(:,2),'k--')
>> axis('square')
>> plot(ss(1,:),ss(2,:),'k--')
>> rect = rand( 7, 4 )

rect =

    0.9501    0.0185    0.1763    0.3529
    0.2311    0.8214    0.4057    0.8132
    0.6068    0.4447    0.9355    0.0099
    0.4860    0.6154    0.9169    0.1389
    0.8913    0.7919    0.4103    0.2028
    0.7621    0.9218    0.8936    0.1987
    0.4565    0.7382    0.0579    0.6038

>> [ur, sr, vr] = svd( rect )

ur =

   -0.2475   -0.0267    0.8328    0.4570   -0.1354    0.0857   -0.0997
   -0.3558   -0.5635   -0.3657    0.4529    0.3123    0.0274   -0.3447
   -0.3711    0.4789   -0.0740    0.2410    0.5619   -0.0149    0.5034
   -0.3911    0.3075   -0.2661    0.2052   -0.5655   -0.5559   -0.1035
   -0.4187   -0.0398    0.2705   -0.6217    0.3422   -0.3738   -0.3264
   -0.5060    0.1892   -0.1440   -0.2468   -0.2640    0.7331   -0.1396
   -0.3003   -0.5661    0.0504   -0.1958   -0.2486   -0.0746    0.6936


sr =

    2.9354         0         0         0
         0    1.0237         0         0
         0         0    0.8440         0
         0         0         0    0.4156
         0         0         0         0
         0         0         0         0
         0         0         0         0


vr =

   -0.5548    0.1316    0.8138   -0.1123
   -0.5867   -0.3284   -0.4300   -0.6025
   -0.5230    0.6023   -0.3905    0.4596
   -0.2730   -0.7156    0.0183    0.6427

>> rect - ur*sr*vr'

ans =

  1.0e-015 *

   -0.3331   -0.3331   -0.5274   -0.4441
    0.1943   -0.1110   -0.2776   -0.1110
    0.1110   -0.1110   -0.3331   -0.0434
    0.0555   -0.1110   -0.4441         0
   -0.1110         0   -0.3886   -0.2220
    0.2220    0.1110   -0.4441   -0.0278
    0.1110         0   -0.4649   -0.2220

>> B

B =

     5     4
     6     5

>> UB

UB =

   -0.6340   -0.7733
   -0.7733    0.6340

>> SB

SB =

   10.0990         0
         0    0.0990

>> VB

VB =

   -0.7733   -0.6340
   -0.6340    0.7733

>> B(:,1)

ans =

     5
     6

>> UB(:,1)

ans =

   -0.6340
   -0.7733

>> VB(:,1)

ans =

   -0.7733
   -0.6340

>> B*VB(:,1)

ans =

   -6.4027
   -7.8100

>> SB*ans

ans =

  -64.6607
   -0.7733

>> UB*ans

ans =

   41.5922
   49.5145

>> B*VB(:,1)

ans =

   -6.4027
   -7.8100

>> SB(1)*UB(:,1)

ans =

   -6.4027
   -7.8100

>> B*VB(:,2)

ans =

   -0.0766
    0.0628

>> SB(2)*UB(:,2)

ans =

     0
     0

>> SB

SB =

   10.0990         0
         0    0.0990

>> SB(2,2)*UB(:,2)

ans =

   -0.0766
    0.0628

>> B*VB(:,2)

ans =

   -0.0766
    0.0628

>> 