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 = {| .362674-3.68203e-17ii  5.87881-3.63087e-14ii .067722+1.54853e-17ii |,
      | -.168148-2.77745e-17ii 10.2268-6.65013e-14ii .359604+2.04246e-17ii | 
      | .287667-1.21253e-17ii  5.41301-3.37944e-14ii .178413+1.81448e-17ii | 
      | .399967-2.6106e-17ii   10.4365-6.58029e-14ii .172408+1.7589e-17ii  | 
      | .893699-1.06778e-18ii  2.90938-1.8657e-14ii  .127632+6.24591e-18ii | 
      | -.108865-2.94833e-17ii 5.44614-3.59076e-14ii .099289+1.64435e-17ii | 
     ------------------------------------------------------------------------
     | .675164+1.30005e-15ii  .770431+2.45685e-15ii -.0909566-8.79201e-16ii 
     | .0675706+9.80659e-16ii .870501+4.49986e-15ii .150312-1.15964e-15ii   
     | .390574+4.28121e-16ii  .658364+2.28673e-15ii -.00751668-1.03019e-15ii
     | .621526+9.2175e-16ii   1.17845+4.45261e-15ii -.0078269-9.98639e-16ii 
     | .902762+3.77011e-17ii  .284467+1.26244e-15ii .0636297-3.5462e-16ii   
     | .141356+1.041e-15ii    .394186+2.42972e-15ii -.0692081-9.33604e-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.