lattice.at

image_lattice

Compute what appears to be called the Hermite form of an integral matrix.

|M| is an $n\times k$ matrix, so columns of |M| span a $\Z$-submodule
$ W=image(M)$ of $\Z^n$; get canonical generators for this submodule

is_saturated

whether |image_lattice(M)| is a saturated lattice (direct factor of $\Z^n$)

adapted_direct_sum

The function |adapted_direct_sum| finds the dimenion |k| of the image of |M|,
and a $\Z$-inverible matrix |R| whose first |c| columns span the saturation of
the image, while remaining ones span a chosen complement of that saturation.

Apart from the initial call of |echelon| to standardise the image and compute
the list of |pivot| positions, this function is very similar to |ech_up| in
matrix.at, but the inverse of the recording matrix is needed here, and since
replacing columns is easier than replacing rows, we maintain the matrix in
inverse transposed form so that we can still maintina it by column operations.

We avoid using |adapted_basis| and |diagonalize|, which may suffer severe
coefficient growth for not too small matrices; we do however use |echelon|
and |Bezout| that might have similar effects. It is to be used in situations
where only the (saturated) image subspace as a whole, and possibly a choice of
a complementary subspace, are needed, not a special basis for the image.

image_subspace

generators for $\Q$-span of image of |M|

image_complement_basis

saturation

intersection of $\Z^n$ with $\Q$-span

free_quotient_lattice_basis

a basis representing the quotient by sublattice spanned by columns of |M|;
tests that this sublattice is a direct factor, and throws an error if not

saturation_quotient_basis

a basis representing the quotient by saturation of sublattice, so no test

projector_to_image

|M| is an $n\times k$ matrix; its columns span a submodule $W$ of $\Z^n$.
Compute $n\times n$ matrix of projection onto the saturation (or $\Q$-span)
of $W$, parallel to its complement from |adapted_direct_sum(M)|.

image_projector

alternate form: an mxc matrix, giving projection to Im(M) in the basis image_subspace(M)

projector_mod_image

complementary projector, to chosen complement to |image_subspace(M)|

mod_image_projector

alternate form: an mx(m-c) matrix, giving projection to the complement of Im(M),
in the basis image_complement_basis(M)

decompose 2 overloads

|M| is an $n\times k$ matrix; its columns span a submodule $W$ of $\Z^n$.
A complement to |image_subspace(M)| is chosen by |adapted_direct_sum(M)|, and
|decompose(M,w)=(u,v)| with |u| in |image_subspace(M)| and |v| in complement.

also defined in character_tables.at

restrict_action

assuming M injective and Im(M) is A-stable, return unique B with A*M=M*B,
which is action of A restricted to that sublattice $\Z^m$, embedded via M

shapes A: n by n, M: n by m

corestrict_action

assuming M full row rank and ker(M) is A-stable, find unique B with M*A=B*M

shapes M: m by n, A: n by n

quotient_matrix

|M| is an $n\times k$ matrix; its columns span a subspace $W$ of $\Q^n$.
Assuming this subspace is |A|-stable, |A| induces an operator in $\Q^n/W$;
return its matrix w.r.t the complement chosen by |adapted_direct_sum(M)|.

solve_ratvec

in_lattice

whether v in image lattice of A

also in this file at line 224

in_lattice_subspace_sum

whether in the sum of lattice |L| and rational subspace spanned by |V|

vec_solve

integer solution to A*x=b

in_lattice_basis

express columns of M relative to basis A of a sub-lattice that contains them
A: nxm matrix, columns are a basis of a Z-module V of rank m
M: nxr matrix, columns span a submodule of V; returns mxr matrix C,
column i gives column i of M as a sum of columns of A i.e., A*C=M

select_columns

select_rows

first_columns

first_rows

intersection_plus

basis of intersection of two matrices
returns (X,Y) where the columns of X span the intersection
Y is the first rows of kernel A##B

intersection

just return the matrix giving the intersection

also defined in sub_cells.at

adapted_direct_sum_plus

adapted_direct_sum in all its glory
A: (r,n) matrix
 E: echelon form of A is an (r,n) matrix
 E= M##Z where M (r,c) matrix (nonzero columns) and Z=(r,n-c) zero matrix
 echelon(A)=(mat M,mat C,vec s,int eps)  (see atlas-functions.help)
 M: (r,c) matrix, C (n,n) invertible  (s: pivot rows, eps is a sign)
 c=#columns of M
 n-c=dimension(kernel(A))
 Then AC=E=M##Z
 Furthermore M -> (mat N,mat R)
 E: (r,c)  N:(r,c)  R: (r,r) invertible
 (RN)##Z=E##Z=AC
 AC=RN##Z
 Note: image_complement_basis is the last n-c columns of R

finite_quotient

generators of finite abelian group Z^n/A

in_lattice

test if ratvec v is in Z-linear span of columns of ratmat A
require: columns of A are linearly independent
let w=unique solution to A*w=v,
return false if w doesn't exist, or boolean is_integer(w) if it does

also in this file at line 136