This page is devoted to answering some basic linear algebra or matrix-theoretic questions along the line "How do I construct ... in GAP?" You may view the html source code for the GAP commands without the output or GAP prompt.

Please send suggestions, additions, corrections to David Joyner.

How do I compute the ... of a matrix? row reduced echelon form transpose determinant characteristic polynomial	How do I find the ... of a group representation? character table Brauer characters
How do I compute ..., where A is a matrix? An mod m (A square) A mod m A*v (v a vector)	How do I compute a basis for the ... of A, where A is a matrix? row span column span image kernel

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• The key command for row reduction is "TriangulizeMat". The following example illustrates the syntax.

  gap> M:=[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
  [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 1, 2, 1 ], [ 1, 1, 0, 0, 0 ] ]
  gap> TriangulizeMat(M);
  gap> M;
  [ [ 1, 0, 0, -1, 1 ], [ 0, 1, 0, 1, -1 ], [ 0, 0, 1, 1, 2 ] ]
  gap> Display(M);
  [ [   1,   0,   0,  -1,   1 ],
    [   0,   1,   0,   1,  -1 ],
    [   0,   0,   1,   1,   2 ] ]
  gap> M:=Z(3)^0*[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]];
  [ [ Z(3)^0, Z(3), 0*Z(3), Z(3)^0, Z(3) ],
    [ Z(3)^0, Z(3), Z(3)^0, Z(3), Z(3)^0 ],
    [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ] ]
  gap> TriangulizeMat(M);
  gap> Display(M);
   1 . . 2 1
   . 1 . 1 2
   . . 1 1 2
  gap>
  

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• There are different methods for matrices over the integers and matrices over a field.

  For integer entries, related commands include "NullspaceIntMat" and "SolutionNullspaceIntMat" in section 25.1 "Linear equations over the integers and Integral Matrices" of the reference manual.

  gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
  [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
  gap> NullspaceIntMat(M);
  [ [ 1, -2, 1 ] ]
  gap> SolutionNullspaceIntMat(M,[0,0,1]);
  [ fail, [ [ 1, -2, 1 ] ] ]
  gap> SolutionNullspaceIntMat(M,[0,0,0]);
  [ [ 0, 0, 0 ], [ [ 1, -2, 1 ] ] ]
  gap> SolutionNullspaceIntMat(M,[1,2,3]);
  [ [ 1, 0, 0 ], [ [ 1, -2, 1 ] ] ]
  
  

  Here (0,0,1) is not in the image of M (under v-> v*M) but (0,0,0) and (1,2,3) are.

  For field entries, related commands include "NullspaceMat" and "TriangulizedNullspaceMat" in section 24.6 "Matrices Representing Linear Equations and the Gaussian Algorithm" of the reference manual.

  gap> M:=[[1,2,3],[4,5,6],[7,8,9]];
  [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ]
  gap> NullspaceMat(M);
  [ [ 1, -2, 1 ] ]
  gap> TriangulizedNullspaceMat(M);
  [ [ 1, -2, 1 ] ]
  gap> M:=[[1,2,3,1,1],[4,5,6,1,1],[7,8,9,1,1],[1,2,3,1,1]];
  [ [ 1, 2, 3, 1, 1 ], [ 4, 5, 6, 1, 1 ], [ 7, 8, 9, 1, 1 ], 
    [ 1, 2, 3, 1, 1 ] ]
  gap> NullspaceMat(M);
  [ [ 1, -2, 1, 0 ], [ -1, 0, 0, 1 ] ]
  gap> TriangulizedNullspaceMat(M);
  [ [ 1, 0, 0, -1 ], [ 0, 1, -1/2, -1/2 ] ]
  
  
  

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• Please see section 24.12.1 of the GAP reference manual for examples of characteristic polynomial of a square matrix ("CharacteristicPolynomial") and section 56.3 for examples of the "characteristic polynomial" (called a "TracePolynomial") of an element of a field extension.

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• GAP contains very extensive character theoretic functions and data libraries (including an interface the character table in the Atlas). Here is just one simple example.

  gap> G:=Group((1,2)(3,4),(1,2,3));
  Group([ (1,2)(3,4), (1,2,3) ])
  gap> T:=CharacterTable(G);
  CharacterTable( Alt( [ 1 .. 4 ] ) )
  gap> Display(T);
  CT1
  
       2  2  2  .  .
       3  1  .  1  1
  
         1a 2a 3a 3b
      2P 1a 1a 3b 3a
      3P 1a 2a 1a 1a
  
  X.1     1  1  1  1
  X.2     1  1  A /A
  X.3     1  1 /A  A
  X.4     3 -1  .  .
  
  A = E(3)^2
    = (-1-ER(-3))/2 = -1-b3
  gap> irr:=Irr(G);
  [ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ),
    Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ),
    Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ),
    Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ]
  gap> Display(irr);
  [ [       1,       1,       1,       1 ],
    [       1,       1,  E(3)^2,    E(3) ],
    [       1,       1,    E(3),  E(3)^2 ],
    [       3,      -1,       0,       0 ] ]
  gap> chi:=irr[2]; gamma:=CG[3]; g:=Representative(gamma); g^chi;
  Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] )
  (1,2,3)^G
  (1,2,3)
  E(3)^2
  
  

  For further details and examples, see chapters 69- 72 of the GAP reference manual.

• Just a simple example of what GAP can do here. To construct a Brauer character table:

  gap> G:=Group((1,2)(3,4),(1,2,3));
  
  Group([ (1,2)(3,4), (1,2,3) ])
  
  gap> irr:=IrreducibleRepresentations(G,GF(7));
  
  [ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ],
    
  [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ],
    
  [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ],
    
  [ (1,2)(3,4), (1,2,3) ] -> [
        
  [ [ 0*Z(7), Z(7)^3, Z(7)^0 ], [ 0*Z(7), Z(7)^3, 0*Z(7) ], 
  [ Z(7)^0, Z(7)^3, 0*Z(7) ] ],
        [ [ 0*Z(7), Z(7)^0, 0*Z(7) ], 
  [ 0*Z(7), 0*Z(7), Z(7)^0 ], [ Z(7)^0, 0*Z(7), 0*Z(7) ] ]
       
  ] ]
  
  gap> brvals := List(irr,chi-> List(ConjugacyClasses(G),c->
  BrauerCharacterValue(Image(chi, Representative(c)))));
  
  [ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ], 
  [ 3, -1, 0, 0 ] ]
  
  gap> Display(brvals);
  
  [ [       1,       1,       1,       1 ],
    
  [       1,       1,  E(3)^2,    E(3) ],
    
  [       1,       1,    E(3),  E(3)^2 ],
    
  [       3,      -1,       0,       0 ] ]
  
  gap>                                               
  
  

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• This too is easy and intuitive:

  gap> A:=[[1,2],[3,4]]; n:=100000; m:=123;
  [ [ 1, 2 ], [ 3, 4 ] ]
  100000
  123
  gap> A^n mod m;
  [ [ 1, 41 ], [ 0, 1 ] ]
  

Last modified 10-14-2005 by David Joyner.