# 35 Integral matrices and lattices

This is a subset of the functions available in GAP4, ported to GAP3 to be used by CHEVIE.

## 35.1 NullspaceIntMat

NullspaceIntMat( mat )

If mat is a matrix with integral entries, this function returns a list of vectors that forms a basis of the integral nullspace of mat, i.e. of those vectors in the nullspace of mat that have integral entries.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
gap> NullspaceMat(mat);
[ [ 1, 0, 3/4, -1/4, -3/4 ], [ 0, 1, -13/24, 1/8, -7/24 ] ]
gap> NullspaceIntMat(mat);
[ [ 1, 18, -9, 2, -6 ], [ 0, 24, -13, 3, -7 ] ] 

## 35.2 SolutionIntMat

SolutionIntMat( mat, vec )

If mat is a matrix with integral entries and vec a vector with integral entries, this function returns a vector x with integer entries that is a solution of the equation x*mat=vec. It returns false if no such vector exists.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
gap> SolutionMat(mat,[95,115,182]);
[ 47/4, -17/2, 67/4, 0, 0 ]
gap> SolutionIntMat(mat,[95,115,182]);
[ 2285, -5854, 4888, -1299, 0 ] 

## 35.3 SolutionNullspaceIntMat

SolutionNullspaceIntMat( mat, vec )

This function returns a list of length two, its first entry being the result of a call to SolutionIntMat with same arguments, the second the result of NullspaceIntMat applied to the matrix mat. The calculation is performed faster than if two separate calls would be used.

    gap> mat:=[[1,2,7],[4,5,6],[7,8,9],[10,11,19],[5,7,12]];;
gap> SolutionNullspaceIntMat(mat,[95,115,182]);
[ [ 2285, -5854, 4888, -1299, 0 ],
[ [ 1, 18, -9, 2, -6 ], [ 0, 24, -13, 3, -7 ] ] ]

## 35.4 BaseIntMat

BaseIntMat( mat )

If mat is a matrix with integral entries, this function returns a list of vectors that forms a basis of the integral row space of mat, i.e. of the set of integral linear combinations of the rows of mat.

    gap> mat:=[[1,2,7],[4,5,6],[10,11,19]];;
gap> BaseIntMat(mat);
[ [ 1, 2, 7 ], [ 0, 3, 7 ], [ 0, 0, 15 ] ]

## 35.5 BaseIntersectionIntMats

BaseIntersectionIntMats( m, n )

If m and n are matrices with integral entries, this function returns a list of vectors that forms a basis of the intersection of the integral row spaces of m and n.

    gap> nat:=[[5,7,2],[4,2,5],[7,1,4]];;
gap> BaseIntMat(nat);
[ [ 1, 1, 15 ], [ 0, 2, 55 ], [ 0, 0, 64 ] ]
gap> BaseIntersectionIntMats(mat,nat);
[ [ 1, 5, 509 ], [ 0, 6, 869 ], [ 0, 0, 960 ] ]

## 35.6 ComplementIntMat

ComplementIntMat( full, sub )

Let full be a list of integer vectors generating an Integral module M and sub a list of vectors defining a submodule S. This function computes a free basis for M that extends S, that is, if the dimension of S is n it determines a basis {b1,...,bm} for M, as well as n integers xi such that xi| xj for i< j and the n vectors si:=xi. bi for i=1,...,n form a basis for S.

It returns a record with the following components:

complement:

the vectors bn+1 up to bm (they generate a complement to S).

sub:

the vectors si (a basis for S).

moduli:

the factors xi.

    gap> m:=IdentityMat(3);;
gap> n:=[[1,2,3],[4,5,6]];;
gap> ComplementIntMat(m,n);
rec( complement := [ [ 0, 0, 1 ] ], sub := [ [ 1, 2, 3 ], [ 0, 3, 6 ] ],
moduli := [ 1, 3 ] ) 

## 35.7 TriangulizedIntegerMat

TriangulizedIntegerMat( mat )

Computes an integral upper triangular form of a matrix with integer entries.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> TriangulizedIntegerMat(m);
[ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]

## 35.8 TriangulizedIntegerMatTransform

TriangulizedIntegerMatTransform( mat )

Computes an integral upper triangular form of a matrix with integer entries. It returns a record with a component normal (a matrix in upper triangular form) and a component rowtrans that gives the transformations done to the original matrix to bring it into upper triangular form.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> n:=TriangulizedIntegerMatTransform(m);
rec( normal := [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ],
rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
rowQ := [ [ 1, 0, 0 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ], rank := 3,
signdet := 1, rowtrans := [ [ 1, 0, 0 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ] )
gap> n.rowtrans*m=n.normal;
true

## 35.9 TriangulizeIntegerMat

TriangulizeIntegerMat( mat )

Changes mat to be in upper triangular form. (The result is the same as that of TriangulizedIntegerMat, but mat will be modified, thus using less memory.)

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> TriangulizeIntegerMat(m); m;
[ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]

## 35.10 HermiteNormalFormIntegerMat

HermiteNormalFormIntegerMat( mat )

This operation computes the Hermite normal form of a matrix mat with integer entries. The Hermite Normal Form (HNF), H of an integer matrix, A is a row equivalent upper triangular form such that all off-diagonal entries are reduced modulo the diagonal entry of the column they are in. There exists a unique unimodular matrix Q such that QA = H.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> HermiteNormalFormIntegerMat(m);
[ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ]

## 35.11 HermiteNormalFormIntegerMatTransform

HermiteNormalFormIntegerMatTransform( mat )

This operation computes the Hermite normal form of a matrix mat with integer entries. It returns a record with components normal (a matrix H of the Hermite normal form) and rowtrans (a unimodular matrix Q) such that Qmat=H

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> n:=HermiteNormalFormIntegerMatTransform(m);
rec( normal := [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ],
rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ], rank := 3,
signdet := 1,
rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ] )
gap> n.rowtrans*m=n.normal;
true

## 35.12 SmithNormalFormIntegerMat

SmithNormalFormIntegerMat( mat )

This operation computes the Smith normal form of a matrix mat with integer entries. The Smith Normal Form,S, of an integer matrix A is the unique equivalent diagonal form with Si dividing Sj for i < j. There exist unimodular integer matrices P, Q such that PAQ = S.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> SmithNormalFormIntegerMat(m);
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]

## 35.13 SmithNormalFormIntegerMatTransforms

SmithNormalFormIntegerMatTransforms( mat )

This operation computes the Smith normal form of a matrix mat with integer entries. It returns a record with components normal (a matrix S), rowtrans (a matrix P), and coltrans (a matrix Q) such that PmatQ=S.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> n:=SmithNormalFormIntegerMatTransforms(m);
rec( normal := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ],
rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
colC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
colQ := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ], rank := 3,
signdet := 1,
rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
coltrans := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ] )
gap> n.rowtrans*m*n.coltrans=n.normal;
true

## 35.14 DiagonalizeIntMat

DiagonalizeIntMat( mat )

This function changes mat to its SNF. (The result is the same as that of SmithNormalFormIntegerMat, but mat will be modified, thus using less memory.)

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> DiagonalizeIntMat(m);m;
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]

## 35.15 NormalFormIntMat

All the previous routines build on the following workhorse' routine: NormalFormIntMat( mat, options ) This general operation for computation of various Normal Forms is probably the most efficient. Options bit values: \begin{itemize} \item{0/1} Triangular Form / Smith Normal Form. \item{2} Reduce off diagonal entries. \item{4} Row Transformations. \item{8} Col Transformations. \item{16} Destructive (the original matrix may be destroyed) \end{itemize} Compute a Triangular, Hermite or Smith form of the n × m integer input matrix A. Optionally, compute n × n and m × m unimodular transforming matrices Q, P which satisfy QA = H or QAP = S. Note option is a value ranging from 0 - 15 but not all options make sense (eg reducing off diagonal entries with SNF option selected already). If an option makes no sense it is ignored. Returns a record with component normal containing the computed normal form and optional components rowtrans and/or coltrans' which hold the respective transformation matrix. Also in the record are components holding the sign of the determinant, signdet, and the Rank of the matrix, rank.

    gap> m:=[[1,15,28],[4,5,6],[7,8,9]];;
gap> NormalFormIntMat(m,0);  # Triangular, no transforms
rec( normal := [ [ 1, 15, 28 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ], rank := 3,
signdet := 1 )
gap> NormalFormIntMat(m,6);  # Hermite Normal Form with row transforms
rec( normal := [ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 3 ] ],
rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ], rank := 3,
signdet := 1,
rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ] )
gap> NormalFormIntMat(m,13); # Smith Normal Form with both transforms
rec( normal := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ],
rowC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
rowQ := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
colC := [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
colQ := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ], rank := 3,
signdet := 1,
rowtrans := [ [ -2, 62, -35 ], [ 1, -30, 17 ], [ -3, 97, -55 ] ],
coltrans := [ [ 1, 0, -1 ], [ 0, 1, -1 ], [ 0, 0, 1 ] ] )
gap> last.rowtrans*m*last.coltrans;
[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 3 ] ]

## 35.16 AbelianInvariantsOfList

AbelianInvariantsOfList( list )

Given a list of positive integers, this routine returns a list of prime powers, such that the prime power factors of the entries in the list are returned in sorted form.

    gap> AbelianInvariantsOfList([4,6,2,12]);
[ 2, 2, 3, 3, 4, 4 ]

## 35.17 Determinant of an integer matrix

DeterminantIntMat( mat )

Computes the determinant of an integer matrix using the same strategy as NormalFormIntMat. This method is faster in general for matrices greater than 20 × 20 but quite a lot slower for smaller matrices. It therefore passes the work to the more general DeterminantMat (see DeterminantMat) for these smaller matrices.

## 35.18 Diaconis-Graham normal form

DiaconisGraham( mat, moduli)

Diaconis and Graham (see dg99) defined a normal form for generating sets of abelian groups. Here moduli should be a list of positive integers such that moduli[i+1] divides moduli[i] for all i, representing the abelian group A=ℤ/moduli×...×ℤ/moduli[n]. The integral matrix m should have n columns where n=Length(moduli), and each line (with the i-th element taken mod moduli[i]) represents an element of the group A.

The function returns false if the set of elements of A represented by the lines of m does not generate A. Otherwise it returns a record r with fields

r.normal:
the Diaconis-Graham normal form, a matrix of same shape as m where either the first n lines are the identity matrix and the remaining lines are 0, or Length(m)=n and .normal differs from the identity matrix only in the entry .normal[n][n], which is prime to moduli[n].

r.rowtrans:
a unimodular matrix such that r.normal=List(r.rowtrans*m,v->Zip(v,moduli, function(x,y)return x mod y;end))

Here is an example:

    gap> DiaconisGraham([[3,0],[4,1]],[10,5]);
rec(
rowtrans := [ [ -13, 10 ], [ 4, -3 ] ],
normal := [ [ 1, 0 ], [ 0, 2 ] ] )
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24 Jun 2022