source: git/Singular/LIB/involut.lib @ 065ddc

version="$Id: involut.lib,v 1.5 2005-05-10 17:31:26 levandov Exp $";
category="Noncommutative";
info="
LIBRARY:  involution.lib  Procedures for Computations and Operations with Involutions
AUTHORS:  Oleksandr Iena,       yena@mathematik.uni-kl.de,
@*        Markus Becker,        mbecker@mathematik.uni-kl.de,
@*        Viktor Levandovskyy,  levandov@mathematik.uni-kl.de

THEORY: Involution is an antiisomorphism of a noncommutative algebra with the
 property that applied an involution twice, one gets an identity. Involution is linear with respect to the ground field. In this library we compute linear involutions, distinguishing the case of a diagonal matrix (such involutions are called homothetic) and a general one.

SUPPORT: Forschungsschwerpunkt 'Mathematik und Praxis' (Project of Dr. E. Zerz
and V. Levandovskyy), Uni Kaiserslautern

NOTE: This library provides algebraic tools for computations and operations
with algebraic involutions and linear automorphisms of noncommutative algebras

PROCEDURES:
find_invo();          describes a variety of linear involutions on a basering;
find_invo_diag();     describes a variety of homothetic (diagonal) involutions on a basering;
find_auto();          describes a variety of linear automorphisms of a basering;
ncdetection();  computes an ideal, presenting an involution map on some particular noncommutative algebras;
involution(m, map theta);  applies the involution to an object.
";

LIB "ncalg.lib";
LIB "poly.lib";
LIB "primdec.lib";
///////////////////////////////////////////////////////////////////////////////
proc ncdetection()
"USAGE:  ncdetection();
RETURN:  ideal, representing an involution map
PURPOSE: compute classical involutions (i.e. acting rather on operators than on variables) for some particular noncommutative algebras
ASSUME: the procedure is aimed at noncommutative algebras with differential, shift or advance operators arising in Control Theory. It has to be executed in the ring.
EXAMPLE: example ncdetection; shows an example
"{
// in this procedure an involution map is generated from the NCRelations
// that will be used in the function involution
// in dieser proc. wird eine matrix erzeugt, die in der i-ten zeile die indices
// der differential-, shift- oder advance-operatoren enthaelt mit denen die i-te
// variable nicht kommutiert.
  if ( nameof(basering)=="basering" )
  {
    "No current ring defined.";
    return(ideal(0));
  }
  def r = basering;
  setring r;
  int i,j,k,LExp;
  int NVars  = nvars(r);
  matrix rel = NCRelations(r)[2];
  intmat M[NVars][3];
  int NRows = nrows(rel);
  intvec v,w;
  poly d,d_lead;
  ideal I;
  map theta;
  for( j=NRows; j>=2; j-- )
  {
   if( rel[j] == w )       //the whole column is zero
    {
      j--;
      continue;
    }
    for( i=1; i<j; i++ )
    {
      if( rel[i,j]==1 )        //relation of type var(j)*var(i) = var(i)*var(j) +1
      {
         M[i,1]=j;
      }
      if( rel[i,j] == -1 )    //relation of type var(i)*var(j) = var(j)*var(i) -1
      {
        M[j,1]=i;
      }
      d = rel[i,j];
      d_lead = lead(d);
      v = leadexp(d_lead); //in the next lines we check wether we have a  relation of differential or shift type
      LExp=0;
      for(k=1; k<=NVars; k++)
      {
        LExp = LExp + v[k];
      }
      //      if( (d-d_lead != 0) || (LExp > 1) )
if ( ( (d-d_lead) != 0) || (LExp > 1) || ( (LExp==0) && ((d_lead>1) || (d_lead<-1)) ) )
      {
        return(theta);
      }

      if( v[j] == 1)                   //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(j)
      {
        if (leadcoef(d) < 0)
        {
          M[i,2] = j;
        }
        else
        {
          M[i,3] = j;
        }
      }
      if( v[i]==1 )                    //relation of type var(j)*var(i) = var(i)*var(j) -lambda*var(i)
      {
        if (leadcoef(d) > 0)
        {
          M[j,2] = i;
        }
        else
        {
          M[j,3] = i;
        }
      }
    }
  }
  // from here on, the map is computed
  for(i=1;i<=NVars;i++)
  {
    I=I+var(i);
  }

  for(i=1;i<=NVars;i++)
  {
    if( M[i,1..3]==(0,0,0) )
    {
      i++;
      continue;
    }
    if( M[i,1]!=0 )
    {
      if( (M[i,2]!=0) && (M[i,3]!=0) )
      {
        I[M[i,1]] = -var(M[i,1]);
        I[M[i,2]] = var(M[i,3]);
        I[M[i,3]] = var(M[i,2]);
      }
      if( (M[i,2]==0) && (M[i,3]==0) )
      {
        I[M[i,1]] = -var(M[i,1]);
      }
      if( ( (M[i,2]!=0) && (M[i,3]==0) )|| ( (M[i,2]!=0) && (M[i,3]==0) )
)
      {
        I[i] = -var(i);
      }
    }
    else
    {
      if( (M[i,2]!=0) && (M[i,3]!=0) )
      {
        I[i] = -var(i);
        I[M[i,2]] = var(M[i,3]);
        I[M[i,3]] = var(M[i,2]);
      }
      else
      {
        I[i] = -var(i);
      }
    }
  }
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r=0,(x,y,z,D(1..3)),dp;
  matrix D[6][6];
  D[1,4]=1; D[2,5]=1;  D[3,6]=1;
  ncalgebra(1,D);
  ncdetection();
  kill r;
  //----------------------------------------
  ring r=0,(x,S),dp;
  ncalgebra(1,-S);
  ncdetection();
  kill r;
  //----------------------------------------
  ring r=0,(x,D(1),S),dp;
  matrix D[3][3];
  D[1,2]=1;  D[1,3]=-S;
  ncalgebra(1,D);
  ncdetection();
}

static proc In_Poly(poly mm, list l, int NVars)
// applies the involution to the poly mm
// entries of a list l are images of variables under invo
// more general than invo_poly; used in many rings setting
{
  int i,j;
  intvec v;
  poly pp, zz;
  poly nn = 0;
  i = 1;
  while(mm[i]!=0)
  {
    v  = leadexp(mm[i]);
    zz = 1;
    for( j=NVars; j>=1; j--)
    {
      if (v[j]!=0)
      {
        pp = l[j];
        zz = zz*(pp^v[j]);
      }
    }
    nn = nn + (leadcoef(mm[i])*zz);
    i++;
  }
  return(nn);
}

static proc Hom_Poly(poly mm, list l, int NVars)
// applies the endomorphism to the poly mm
// entries of a list l are images of variables under endo
// should not be replaced by map-based stuff! used in
// many rings setting
{
  int i,j;
  intvec v;
  poly pp, zz;
  poly nn = 0;
  i = 1;
  while(mm[i]!=0)
  {
    v  = leadexp(mm[i]);
    zz = 1;
    for( j=NVars; j>=1; j--)
    {
      if (v[j]!=0)
      {
        pp = l[j];
        zz = (pp^v[j])*zz;
      }
    }
    nn = nn + (leadcoef(mm[i])*zz);
    i++;
  }
  return(nn);
}

static proc invo_poly(poly m, map theta)
// applies the involution map theta to m, where m=polynomial
{
  // compatibility:
  ideal l = ideal(theta);
  int i;
  list L;
  for (i=1; i<=size(l); i++)
  {
    L[i] = l[i];
  }
  int nv = nvars(basering);
  return (In_Poly(m,L,nv));
//   if (m==0) { return(m); }
//   int i,j;
//   intvec v;
//   poly p,z;
//   poly n = 0;
//   i = 1;
//   while(m[i]!=0)
//   {
//     v = leadexp(m[i]);
//     z =1;
//     for(j=nvars(basering); j>=1; j--)
//     {
//       if (v[j]!=0)
//       {
//         p = var(j);
//         p = theta(p);
//         z = z*(p^v[j]);
//       }
//     }
//     n = n + (leadcoef(m[i])*z);
//     i++;
//   }
//   return(n);
}
///////////////////////////////////////////////////////////////////////////////////
proc involution(m, map theta)
"USAGE:  involution(m, theta); m is a poly/vector/ideal/matrix/module, theta is a map
RETURN:  object of the same type as m
PURPOSE: applies the involution, presented by theta to the object m
THEORY: for an involution theta and two polynomials a,b from the algebra, theta(ab) = theta(b) theta(a); theta is linear with respect to the ground field
EXAMPLE: example involution; shows an example
"{
  // applies the involution map theta to m,
  // where m= vector, polynomial, module, matrix, ideal
  int i,j;
  intvec v;
  poly p,z;
  if (typeof(m)=="poly")
  {
    return (invo_poly(m,theta));
  }
  if ( typeof(m)=="ideal" )
  {
    ideal n;
    for (i=1; i<=size(m); i++)
    {
      n[i] = invo_poly(m[i], theta);
    }
    return(n);
  }
  if (typeof(m)=="vector")
  {
    for(i=1; i<=size(m); i++)
    {
      m[i] = invo_poly(m[i], theta);
    }
    return (m);
  }
  if ( (typeof(m)=="matrix") || (typeof(m)=="module"))
  {
    matrix n = matrix(m);
    int @R=nrows(n);
    int @C=ncols(n);
    for(i=1; i<=@R; i++)
    {
      for(j=1; j<=@C; j++)
      {
        if (m[i,j]!=0)
        {
          n[i,j] = invo_poly( m[i,j], theta);
        }
      }
    }
    if (typeof(m)=="module")
    {
      return (module(n));
    }
    else // matrix
    {
      return(n);
    }
  }
  // if m is not of the supported type:
  "Error: unsupported argument type!";
  return();
}
example
{
  "EXAMPLE:";echo = 2;
  ring r = 0,(x,d),dp;
  ncalgebra(1,1); // Weyl-Algebra
  map F = r,x,-d;
  poly f =  x*d^2+d;
  poly If = involution(f,F);
  f-If;
  poly g = x^2*d+2*x*d+3*x+7*d;
  poly tg = -d*x^2-2*d*x+3*x-7*d;
  poly Ig = involution(g,F);
  tg-Ig;
  ideal I = f,g;
  ideal II = involution(I,F);
  II;
  I - involution(II,F);
  module M  = [f,g,0],[g,0,x^2*d];
  module IM = involution(M,F);
  print(IM);
  print(M - involution(IM,F));
}
///////////////////////////////////////////////////////////////////////////////////
static proc new_var()
//generates a string of new variables
{

  int NVars=nvars(basering);
  int i,j;
  string s="@_1_1";
  for(i=1; i<=NVars; i++)
  {
    for(j=1; j<=NVars; j++)
    {
      if(i*j!=1)
      {
        s = s+ ","+NVAR(i,j);
      };
    };
  };
  return(s);
};

static proc NVAR(int i, int j)
{
  return("@_"+string(i)+"_"+string(j));
};
///////////////////////////////////////////////////////////////////////////////////
static proc new_var_special()
//generates a string of new variables
{
  int NVars=nvars(basering);
  int i;
  string s="@_1_1";
  for(i=2; i<=NVars; i++)
  {
    s = s+ ","+NVAR(i,i);
  };
  return(s);
};
///////////////////////////////////////////////////////////////////////////////////
static proc RelMatr()
// returns the matrix of relations
// only Lie-type relations x_j x_i= x_i x_j + .. are taken into account
{
  int i,j;
  int NVars = nvars(basering);
  matrix Rel[NVars][NVars];
  for(i=1; i<NVars; i++)
  {
    for(j=i+1; j<=NVars; j++)
    {
      Rel[i,j]=var(j)*var(i)-var(i)*var(j);
    };
  };
  return(Rel);
};
/////////////////////////////////////////////////////////////////
proc find_invo()
"USAGE: find_invo();
RETURN: a ring containing a list L of pairs, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]

PURPOSE: computed the ideal of linear involutions of the basering

NOTE: for convenience, the full ideal of relations @code{idJ}
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
EXAMPLE: example find_invo; shows examples
SEE ALSO: find_invo_diag, involution
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  matrix Rel = RelMatr(); //the matrix of relations

  string s   = new_var(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring

  if (Par=="") // if there are no parameters
  {
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
     execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };

  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring

  int Sz = NVars*NVars+NVars; // number of variables in new ring

  matrix M[Sz][Sz]; //to be the matrix of relations in new ring

  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };

  ncalgebra(1, M); //now new ring @@K become a noncommutative ring

  list l; //list to define an involution
  poly @@F;
  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    @@F=0;
    for(j=1; j<=NVars; j++)
    {
      execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" );
    };
    l=l+list(@@F);
  };

  matrix N = Rel; //imap(@B,Rel);

  for(i=1; i<NVars; i++)//get matrix by applying the involution to relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N
  ideal J;
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) );
  };
  J = simplify(J,2);
  //-------------------------------------------------
  if ( Par=="" ) //initializes the ring of relations
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables

  J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity
  J = simplify(J,2); // without extra zeros
  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL);
  option(redSB);       // important for reduced GBs
  option(redTail);
  matrix IM = @@D;     // involution map
  list L    = list();  // the answer
  list TL;
  ideal tmp = 0;
  for (i=1; i<=sL; i++) // compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp, NVars,NVars);
    L[i]  = TL;
  }
  export(L); // main export
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_invo();
 setring X; // ring with new variables, corresponding to unknown coefficients
 L;
 print(L[1][2]);  // L[i][2] is a matrix in new variables, defining the linear involution
 L[1][1];  // where new variables obey these relations
}
///////////////////////////////////////////////////////////////////////////
proc find_invo_diag()
"USAGE: find_invo_diag();
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]

PURPOSE: compute the ideal of homothetic (diagonal) involutions of the basering

NOTE: for convenience, the full ideal of relations @code{idJ}
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
EXAMPLE: example find_invo_diag; shows examples
SEE ALSO: find_invo, involution
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  matrix Rel = RelMatr(); //the matrix of relations

  string s   = new_var_special(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring

  if (Par=="") // if there are no parameters
  {
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
    execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };

  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring

  int Sz = 2*NVars; // number of variables in new ring

  matrix M[Sz][Sz]; //to be the matrix of relations in new ring
  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };

  ncalgebra(1, M); //now new ring @@K become a noncommutative ring

  list l; //list to define an involution

  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    execute( "l["+string(i)+"]="+NVAR(i,i)+"*"+string( var(i) )+";" );

  };
  matrix N = Rel; //imap(@B,Rel);

  for(i=1; i<NVars; i++)//get matrix by applying the involution to relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] + In_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N

  ideal J;
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) );
  };
  J = simplify(J,2);
  //-------------------------------------------------

  if ( Par=="" ) //initializes the ring of relations
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };

  ideal J = imap(@@K,J); // ideal, considered in @@KK now

  matrix @@D[NVars][NVars]; // matrix with entries=new variables to square i.e. @@D=@@D^2
  for(i=1;i<=NVars;i++)
  {
    execute("@@D["+string(i)+","+string(i)+"]="+NVAR(i,i)+";");
  };
  J = J, ideal( @@D*@@D - matrix( freemodule(NVars) ) ); // add the condition that involution to square is just identity
  J = simplify(J,2); // without extra zeros

  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL);
  option(redSB); // important for reduced GBs
  option(redTail);
  matrix IM = @@D; // involution map
  list L = list(); // the answer
  list TL;
  ideal tmp = 0;
  for (i=1; i<=sL; i++) // compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp, NVars,NVars);
    L[i]  = TL;
  }
  export(L);
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_invo_diag();
 setring X; // ring with new variables, corresponding to unknown coefficients
 print(L[1][2]);  // a first matrix, defining the linear involution: we see it is constant
 print(L[2][2]);  // and a second possible matrix; it is constant too
 L; // let us take a look on the whole list
}
/////////////////////////////////////////////////////////////////////
proc find_auto()
"USAGE: find_auto();
RETURN: a ring together with a list of pairs L, where
@*        L[i][1]  =  Groebner Basis of an i-th associated prime,
@*        L[i][2]  =  matrix, defining a linear map, with entries, reduced with respect to L[i][1]

PURPOSE: computes the ideal of linear automorphisms of the basering

NOTE: for convenience, the full ideal of relations @code{idJ}
and the initial matrix with indeterminates @code{matD} are exported in the output ring.
EXAMPLE: example find_auto; shows examples
SEE ALSO: find_invo
"{
  def @B    = basering; //save the name of basering
  int NVars = nvars(@B); //number of variables in basering
  int i, j;

  matrix Rel = RelMatr(); //the matrix of relations

  string s   = new_var(); //string of new variables
  string Par = parstr(@B); //string of parameters in old ring

  if (Par=="") // if there are no parameters
  {
    execute("ring @@K=0,("+varstr(@B)+","+s+"), dp;"); //new ring with new variables
  }
  else //if there exist parameters
  {
     execute("ring @@K=(0,"+Par+") ,("+varstr(@B)+","+s+"), dp;");//new ring with new variables
  };

  matrix Rel = imap(@B, Rel); //consider the matrix of relations in new ring

  int Sz = NVars*NVars+NVars; // number of variables in new ring

  matrix M[Sz][Sz]; //to be the matrix of relations in new ring

  for(i=1; i<NVars; i++) //initialize that matrix of relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      M[i,j] = Rel[i,j];
    };
  };

  ncalgebra(1, M); //now new ring @@K become a noncommutative ring

  list l; //list to define a homomorphism(isomorphism)
  poly @@F;
  for(i=1; i<=NVars; i++) //initializing list for involution
  {
    @@F=0;
    for(j=1; j<=NVars; j++)
    {
      execute( "@@F = @@F+"+NVAR(i,j)+"*"+string( var(j) )+";" );
    };
    l=l+list(@@F);
  };

  matrix N = Rel; //imap(@B,Rel);

  for(i=1; i<NVars; i++)//get matrix by applying the homomorphism  to relations
  {
    for(j=i+1; j<=NVars; j++)
    {
      N[i,j]= l[j]*l[i] - l[i]*l[j] - Hom_Poly( N[i,j], l, NVars);
    };
  };
  kill l;
  //---------------------------------------------
  //get the ideal of coefficients of N
  ideal J;
  ideal idN = simplify(ideal(N),2);
  J = ideal(coeffs( idN, var(1) ) );
  for(i=2; i<=NVars; i++)
  {
    J = ideal( coeffs( J, var(i) ) );
  };
  J = simplify(J,2);
  //-------------------------------------------------
  if ( Par=="" ) //initializes the ring of relations
  {
    execute("ring @@KK=0,("+s+"), dp;");
  }
  else
  {
    execute("ring @@KK=(0,"+Par+"),("+s+"), dp;");
  };
  ideal J = imap(@@K,J); // ideal, considered in @@KK now
  string snv = "["+string(NVars)+"]";
  execute("matrix @@D"+snv+snv+"="+s+";"); // matrix with entries=new variables

  J = J, ideal( @@D*@@D-matrix( freemodule(NVars) ) ); // add the condition that homomorphism to square is just identity
  J       = simplify(J,2); // without extra zeros
  list mL = minAssGTZ(J); // components not in GB
  int sL  = size(mL);
  option(redSB); // important for reduced GBs
  option(redTail);
  matrix IM = @@D; //  map
  list L = list(); // the answer
  list TL;
  ideal tmp = 0;
  for (i=1; i<=sL; i++)// compute GBs of components
  {
    TL    = list();
    TL[1] = std(mL[i]);
    tmp   = NF( ideal(IM), TL[1] );
    TL[2] = matrix(tmp,NVars, NVars);
    L[i]  = TL;
  }
  export(L);
  ideal idJ = J; // debug-comfortable exports
  matrix matD = @@D;
  export(idJ);
  export(matD);
  return(@@KK);
}
example
{ "EXAMPLE:"; echo = 2;
 def a = CreateWeyl(1);
 setring a; // this algebra is a first Weyl algebra
 def X = find_auto();
 setring X; // ring with new variables, corresponding to unknown coefficients
 print(L[1][2]);  // a first matrix, defining the linear automorphism : we see it is constant
 print(L[2][2]);  // and a second possible matrix; it is constant too
 L; // let us take a look on the whole list
}