solutionsToAffineCoords -- writes solutions in global coords to affine coordinates.

Synopsis

Usage:

S' = solutionsToAffineCoords S
Inputs:
- S, a list, solutions to a Schubert problem in global coordinates
Outputs:
- S', a list, solutions in coordinates of an affine patch with an identity in the bottom

Description

Takes a list of solutions $s\in Gr(k,n)$ in global coordinates and writes them as solutions in coordinates of the affine patch that has an identity matrix in the last $k$ rows.

i1 : Pblm = {
         ({2,1}, random(RR^6,RR^6)),
         ({2,1}, random(RR^6,RR^6)),
         ({2,1}, random(RR^6,RR^6))
         }

o1 = {({2, 1}, | .892712  .89189  .0741835 .741046 .830833 .191734  |), ({2,
               | .673395  .231053 .808694  .108386 .538554 .403215  |       
               | .29398   .461944 .362835  .348931 .873665 .615911  |       
               | .632944  .775187 .706096  .562428 .415912 .0147867 |       
               | .0258884 .909047 .127435  .246268 .606588 .223028  |       
               | .714827  .314897 .254482  .153346 .848005 .388829  |       
     ------------------------------------------------------------------------
     1}, | .557119 .96518  .0647412 .174853 .444183 .184779 |), ({2, 1}, |
         | .873708 .681683 .877846  .626892 .644366 .370833 |            |
         | .7037   .914199 .0340514 .350611 .194945 .339222 |            |
         | .681869 .887381 .507989  .379495 .518585 .062212 |            |
         | .276259 .169813 .150294  .237252 .987173 .465736 |            |
         | .605659 .965004 .656391  .116721 .568273 .40273  |            |
     ------------------------------------------------------------------------
     .164647 .562839  .0645275 .501243 .205375  .0821679 |)}
     .713493 .629991  .283709  .154289 .276652  .10394   |
     .909537 .479826  .977573  .467203 .0958269 .280679  |
     .566034 .815167  .212436  .765564 .605398  .475179  |
     .305423 .97723   .592747  .305946 .883549  .130004  |
     .732358 .0595849 .831802  .53632  .942865  .522285  |

o1 : List

i2 : S = solveSchubertProblem(Pblm, 3,6)

o2 = {| .362674+4.83909e-16ii  5.87881+1.4973e-13ii  .067722-3.25474e-16ii |,
      | -.168148+3.65025e-16ii 10.2268+2.74238e-13ii .359604-4.29289e-16ii | 
      | .287667+1.59357e-16ii  5.41301+1.39362e-13ii .178413-3.8137e-16ii  | 
      | .399967+3.43098e-16ii  10.4365+2.71358e-13ii .172408-3.69689e-16ii | 
      | .893699+1.40332e-17ii  2.90938+7.69377e-14ii .127632-1.31278e-16ii | 
      | -.108865+3.87484e-16ii 5.44614+1.48076e-13ii .099289-3.45613e-16ii | 
     ------------------------------------------------------------------------
     | .675164+1.12445e-15ii  .770431+2.17408e-15ii -.0909566-7.64252e-16ii 
     | .0675706+8.48201e-16ii .870501+3.98194e-15ii .150312-1.00802e-15ii   
     | .390574+3.70294e-16ii  .658364+2.02353e-15ii -.00751668-8.95503e-16ii
     | .621526+7.9725e-16ii   1.17845+3.94013e-15ii -.0078269-8.68074e-16ii 
     | .902762+3.26088e-17ii  .284467+1.11714e-15ii .0636297-3.08256e-16ii  
     | .141356+9.00388e-16ii  .394186+2.15007e-15ii -.0692081-8.11541e-16ii 
     ------------------------------------------------------------------------
     |}
     |
     |
     |
     |
     |

o2 : List

i3 : solutionsToAffineCoords S

o3 = {| 1.17594  -.247384 -1.04186 |, | -.0717886 .515973  1.79675  |}
      | -2.59821 1.69933  5.94898  |  | 1.97727   -.796664 -3.12795 |
      | -.743029 .890962  1.94183  |  | .504284   .067644  .113771  |
      | 1        0        0        |  | 1         0        0        |
      | 0        1        0        |  | 0         1        0        |
      | 0        0        1        |  | 0         0        1        |

o3 : List

Caveat

This function may fail if the solutions are not in general position (if they cannot fit the specific local coordinates) One way to avoid this is by applying a random linear transformation to the solutions before calling this function

Ways to use solutionsToAffineCoords :

solutionsToAffineCoords(List)

For the programmer

The object solutionsToAffineCoords is a method function.