Matrices and Modules

Since \(S_p^\nu\) and \(S_u^\nu\) are Euclidean, we can work with canonical forms and modules over these rings. We consider \(S_p^\nu\) first.

Matrices

The row-echelon form of a matrix over \(S_p^\nu\) has pivot entries which are polynomials of the form

\[u^d+\sum_{j=0}^{d-1} b_ju^j\qquad{\rm where}\qquad v(b_j)+\nu j >\nu d\]

for \(j\) in the sum, so that the Weierstrass degree is \(d\) and the Gauss valuation is \(d\nu\).

The Hermite form additionally assures that the other entries in the columns with pivots are polynomials of smaller degree, via taking Quotrem. This can be made unique via suitable choices, and so provides a way of testing equality of modules.

The Smith form is computed by creating a matrix of pivots corresponding to a Hermite on the matrix and then a Hermite on the transpose. One also demands recursive divisibility of the resulting diagonal terms.

Over \(S_u^\nu\) the picture is the same, except that pivot entries of the echelon form are all CanonicalElements, with similar changes thereon. E.g., the Hermite form will have elements of smaller valuation above the pivots.

SpMatrixSpace(S, r, c): SpRng, RngIntElt, RngIntElt -> SpMatRng

SuMatrixSpace(S, r, c): SuRng, RngIntElt, RngIntElt -> SuMatRng

Given an \(S_p^\nu\) of \(S^\nu_u\) ring and a (nonnegative) number of rows and columns, construct the matrix space.

IdentityMatrix(S, n): SpRng, RngIntElt -> SpMat

IdentityMatrix(S): SpMatRng -> SpMat

ZeroMatrix(S): SpMatRng -> SpMat

ZeroMatrix(S, n): SpRng, RngIntElt -> SpMat

ZeroMatrix(S, r, c): SpRng, RngIntElt, RngIntElt -> SpMat

IdentityMatrix(S, n): SuRng, RngIntElt -> SuMat

IdentityMatrix(S): SuMatRng -> SuMat

ZeroMatrix(S): SuMatRng -> SuMat

ZeroMatrix(S, n): SuRng, RngIntElt -> SuMat

ZeroMatrix(S, r, c): SuRng, RngIntElt, RngIntElt -> SuMat

Different ways of obtaining the zero matrix or identity matrix (if applicable) of a matrix ring.

SpMatrix(A): SeqEnum -> SpMat

SpMatrix(r, c, A): RngIntElt, RngIntElt, SeqEnum -> SpMat

SpMatrix(v): SeqEnum -> SpMat

SpMatrix(A): SeqEnum -> SpMat

SuMatrix(r, c, A): RngIntElt, RngIntElt, SeqEnum -> SuMat

SuMatrix(v): SeqEnum -> SuMat

Various ways of obtaining an SpMat or SuMat. The first takes a sequence of sequences of SpElements or SuElements, the second takes a row and column count and a sequence of SpElements or SuElements, and the third takes a sequence of SpVecs or SuVecs.

GaussValuations(M): SpMat -> SeqEnum

GaussValuations(M): SuMat -> SeqEnum

Returns a sequence of sequences of the Gauss valuations of the elements of a matrix. Since these can be ExtendedReals, the valuations themselves are put in an array rather than a matrix.

WeierstrassDegrees(M): SpMat -> SeqEnum

WeierstrassDegrees(M): SuMat -> SeqEnum

Returns a sequence of sequences of the Weierstrass degrees of the elements of a matrix. Since these can be ExtendedReals, the valuations themselves are put in an array rather than a matrix.

LeadingTerms(M): SpMat -> Mtrx

LeadingTerms(M): SuMat -> Mtrx

Returns a matrix (defined over the underlying power or Laurent series ring) which has the leading term of each element.

WeierstrassTerms(M): SpMat -> Mtrx

WeierstrassTerms(M): SuMat -> Mtrx

Returns a matrix (defined over the underlying power or Laurent series ring) which has the Weierstrass term of each element.

IsWeaklyZero(M): SpMat -> BoolElt

IsWeaklyZero(M): SuMat -> BoolElt

Returns whether every element in the matrix is nearly zero.

EchelonForm(M): SpMat -> SpMat, SpMat

EchelonForm(M): SuMat -> SuMat, SuMat

Transform: BoolElt                    Default: false

Returns the (row) echelon form \(E\), and (if desired) the transform \(T\) such that \(E=TM\).

HermiteForm(M): SpMat -> SpMat, SpMat

HermiteForm(M): SuMat -> SuMat, SuMat

Transform: BoolElt                    Default: false

Returns the Hermite form \(H\), and (if desired) the transform \(T\) such that \(H=TM\).

SmithForm(M): SpMat -> SpMat, SpMat, SpMat

SmithForm(M): SuMat -> SuMat, SuMat, SuMat

Returns the Smith form \(S\), and (if desired) the transforms \(P,Q\) such that \(S=PMQ\).

Kernel(M): SpMat -> SpSpc

Kernel(M): SuMat -> SuSpc

Image(M): SpMat -> SpSpc

Image(M): SuMat -> SuSpc

Return the kernel or image of an \(S_p^\nu\) or \(S_u^\nu\) matrix as a module space. The Image also has a Transform parameter that gives the inclusion.

Example: Matrices over Sp Rings (ex-8c19ad)

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Here are some basic examples of operations with matrices over \(S^\nu_p\).

> S<u> := SpRing (pAdicField (3, 20), 1/2); // slope 1/2
> M := SpMatrix (2, 3, [ S | u^2/3,u/3^2,0, u^3,3^2*u,3*u ]);
> E, T := EchelonForm (M : Transform); assert E eq T*M;
> LeadingTerms (E);
[(3^-1 + O(3^19))*$.1^2  (3^-2 + O(3^18))*$.1  O($.1^20)]
[O($.1^20)               (3^2 + O(3^18))*$.1   (3 + O(3^21))*$.1]
> WeierstrassTerms (E); // not the same
[(3^-1 + O(3^19))*$.1^2  (3^-2 + O(3^18))*$.1    O($.1^20)]
[O($.1^20)              -(3^-1 + O(3^19))*$.1^2  (3 + O(3^21))*$.1]

> S<u> := SpRing (pAdicField (5, 10), 3/2); // slope 3/2
> M := SpMatrix (3,5,[S|5^3,u,u^2/5,0,0, 0,0,25,u^3/25,3*5, 0,0,0,0,5^2]);
> H, T := HermiteForm (M : Transform); assert H eq T*M;
> WeierstrassTerms(H);
[5^3 + O(5^13)  $.1        O($.1^20)     -(5^-5 + O(5^5))*$.1^5  O($.1^20)]
[O($.1^20)      O($.1^20)  5^2 + O(5^9)  (5^-2 + O(5^8))*$.1^3   O($.1^20)]
[O($.1^20)      O($.1^20)  O($.1^20)     O($.1^20)               5^2 + O(5^12)]
> X, P, Q := SmithForm (M : Transform); assert P*M*Q eq X;
> WeierstrassTerms(X); // note that 5^3 divides 5^2 in this ring!
[5^3 + O(5^13)  O($.1^20)     O($.1^20)      O($.1^20)  O($.1^20)]
[O($.1^20)      5^2 + O(5^9)  O($.1^20)      O($.1^20)  O($.1^20)]
[O($.1^20)      O($.1^20)     5^2 + O(5^12)  O($.1^20)  O($.1^20)]

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Example: Matrices over Su Rings (ex-970ed4)

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Here are similar examples over \(S^\nu_u\).

> S<u> := SuRing (pAdicField (5, 15), 2/3); // slope 2/3
> M := SuMatrix (2, 3, [ S | 5^2/u,0,u^2, u,5*u^2,5^3/u^2 ]);
> E, T := EchelonForm (M : Transform); assert E eq T*M;
> LeadingTerms (E);
[$.1        (5 + O(5^16))*$.1^2   (5^3 + O(5^18))*$.1^-2]
[O($.1^20)  5^3 + O(5^15)         (5^5 + O(5^20))*$.1^-4]
> WeierstrassTerms (E); // not the same, in lower right
[$.1        (5 + O(5^16))*$.1^2  (5^3 + O(5^18))*$.1^-2]
[O($.1^20)  5^3 + O(5^15)        -$.1^2]

> A := [S | 25/u,u,u^2/5,0,0, 0,0,25,125/u^2,50*u, 0,0,0,0,5*u^2];
> M := SuMatrix (3, 5, A);
> H, T := HermiteForm (M : Transform); assert H eq T*M;
> W := WeierstrassTerms(H); W;
[$.1^2 (5^-2 + O(5^13))*$.1^4 (5^-3 + O(5^12))*$.1^5 O($.1^20) O($.1^20)]
[O($.1^20) O($.1^20) 5^2 + O(5^17) (5^3 + O(5^18))*$.1^-2 O($.1^20)]
[O($.1^20) O($.1^20) O($.1^20) O($.1^20) (5 + O(5^16))*$.1^2]
> assert CanonicalElement (S, 4/3) eq S!W[1,1];
> assert CanonicalElement (S, 2) eq S!W[2,3];
> assert CanonicalElement (S, 7/3) eq S!W[3,5];
> X, P, Q := SmithForm (M : Transform); assert X eq P*M*Q;
> WeierstrassTerms(X);
[(5^-1 + O(5^14))*$.1^2 O($.1^20) O($.1^20) O($.1^20) O($.1^20)]
[O($.1^20) (5 + O(5^16))*$.1 O($.1^20) O($.1^20) O($.1^20)]
[O($.1^20) O($.1^20) (5 + O(5^16))*$.1^2 O($.1^20) O($.1^20)]

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Modules

These are implemented as “vectors” over \(S^\nu_p\) or \(S^\nu_u\).

SpSpace(R, n): SpRng, RngIntElt -> SpSpc

SuSpace(R, n): SuRng, RngIntElt -> SuSpc

Given an \(S^\nu_p\) or \(S^\nu_u\) ring and a nonnegative integer, construct the module of that degree.

SpSpace(M): SpMat -> SpSpc

SuSpace(M): SuMat -> SuSpc

Given an \(S^\nu_p\) or \(S^\nu_u\) matrix, construct the associated module space. This always computes the Hermite form to use as the basis.

SpSpace(v): SeqEnum -> SpSpc

SuSpace(v): SeqEnum -> SuSpc

Given a sequence of SpVecs or SuVecs, construct the module space generated by them. Again this always computes the Hermite form for the basis.

Ambient(S): SpSpc -> SpSpc

Ambient(S): SuSpc -> SuSpc

Given a module space, return the (full) ambient space of dimension equal to the degree.

Parent(v): SpVec -> SpSpc

Parent(v): SuVec -> SuSpc

The Parent of an SpVec or SuVec is always its Ambient space.

ZeroVector(S): SpSpc -> SpVec

ZeroVector(S): SuSpc -> SuVec

Return the zero vector of a module space.

SpVector(e): SeqEnum -> SpVec

SuVector(e): SeqEnum -> SuVec

Given a sequence of \(S^\nu_p\) or \(S^\nu_u\) elements, construct the vector of them.

Basis(S): SpSpc -> SeqEnum

Basis(S): SuSpc -> SeqEnum

Given a module space, return a basis as a sequence of vectors.

BasisMatrix(S): SpSpc -> SpMat

BasisMatrix(S): SuSpc -> SuMat

Given a module space, return a basis as a matrix.

Dimension(S): SpSpc -> RngIntElt

Dimension(S): SuSpc -> RngIntElt

Degree(S): SpSpc -> RngIntElt

Degree(S): SuSpc -> RngIntElt

The dimension and degree of a module space.

LeadingTerms(v): SpVec -> ModTupRngElt

LeadingTerms(v): SuVec -> ModTupRngElt

The leading term of each element in an SpVec or SuVec.

WeierstrassTerms(v): SpVec -> ModTupRngElt

WeierstrassTerms(v): SuVec -> ModTupRngElt

The Weierstrass term of each element in an SpVec or SuVec.

GaussValuations(v): SpVec -> SeqEnum

GaussValuations(v): SuVec -> SeqEnum

Returns a sequence of sequences of the Gauss valuations of the elements of a matrix. Since these can be ExtendedReals, the valuations themselves are put in an array rather than a vector.

WeierstrassDegrees(v): SpVec -> SeqEnum

WeierstrassDegrees(v): SuVec -> SeqEnum

Returns a sequence of sequences of the Weierstrass degrees of the elements of a vector. Since these can be ExtendedReals, the valuations themselves are put in an array rather than a vector.

IsWeaklyZero(v): SpVec -> BoolElt

IsWeaklyZero(v): SuVec -> BoolElt

Return whether an SpVec or SuVec is weakly zero.

M1 + M2: SpSpc, SpSpc -> SpSpc

M1 + M2: SuSpc, SuSpc -> SuSpc

DirectSum(M1, M2): SpSpc, SpSpc -> SpSpc

DirectSum(M1, M2): SuSpc, SuSpc -> SuSpc

M1 meet M2: SpSpc, SpSpc -> SpSpc

M1 meet M2: SuSpc, SuSpc -> SuSpc

The sum and intersection of two module spaces. The use of DirectSum requires the summands to be disjoint.

S * M: SpSpc, SpMat -> SpSpc

S * M: SuSpc, SuMat -> SuSpc

The transformation of a module space by a matrix. The basis of the resulting space will be given in Hermite form.

IsConsistent(M, v): SpMat, SpVec -> BoolElt, SpVec

IsConsistent(M, v): SuMat, SuVec -> BoolElt, SuVec

IsConsistent(M, e): SpMat, SeqEnum -> BoolElt, SeqEnum

IsConsistent(M, e): SuMat, SeqEnum -> BoolElt, SeqEnum

IsConsistent(M, W): SpMat, SpMat -> BoolElt, SpMat, SpMat

IsConsistent(M, W): SuMat, SpMat -> BoolElt, SuMat, SuMat

v in M: SpVec, SpMat -> BoolElt, SpVec

v in M: SuVec, SpMat -> BoolElt, SuVec

Determines whether \(xM=v\) is solvable, and it so returns such an \(x\). The second version takes an a sequence of vectors, and the third takes a matrix (and also includes the kernel as a third return value).

IsSubspace(A, B): SpSpc, SpSpc -> BoolElt, SpMat

IsSubspace(A, B): SuSpc, SuSpc -> BoolElt, SuMat

Determines whether \(A\) is a subspace of \(B\), and if so gives an inclusion map. Equality can be determined via symmetric inclusion, or alternatively via the Hermite form (which should be unique) of the basis matrix.

Example: Modules over Sp Rings (ex-3f4950)

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Here are some basic examples of usage for modules.

> S<u> := SpRing (pAdicField (5, 25), 1/2); // slope 1/2
> A := SpSpace (S, 3);
> A.1;
[ 1 + O(5^25) + O(u^20), O(u^20), O(u^20) ]
> M := SpMatrix (2, 3, [ S | 5*u,5,u^2, 1/5,u^3,u ]);
> B := SpSpace (M); assert Dimension(B) eq 2;
> LeadingTerms(BasisMatrix(B));
[5^-1 + O(5^24) O($.1^20)    $.1]
[O($.1^20)      -5 + O(5^26) (24 + O(5^25))*$.1^2]
> B eq SpSpace (VerticalJoin (M, M)); // also gives inclusion
true
> A meet B eq B;
true
> Image(M) eq B; // by definition
true
> IsConsistent (M, A.1);
false
> IsConsistent (M, B.1);
true
> SpSpace([B.1]) + SpSpace([B.2]) eq B;
true
> SpSpace([B.1]) + SpSpace([u * B.2]) eq B; // u does not invert
false
> SpSpace([1/5 * B.1]) + SpSpace([5 * B.2]) eq B;
true
> sub<B|[B.1+B.2]> + sub<B|[B.2-B.1]> eq sub<B|[B.1,B.2]>;
true

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Example: Module with a Nontrivial Kernel (ex-76e436)

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Here is an example over \(S^\nu_p\) with a nontrivial kernel.

> S<u> := SpRing (pAdicField (7, 30), 1); // slope 1
> M := SpMatrix(3,5,[S|1,0,u,u^2,u,  u^2/7,0,1,0,0, 0,0,1,0,u^4/7^2]);
> ROWS := Rows(M);
> V := u * ROWS[1] + u^2/7 * ROWS[3]; // create a kernel
> MAT := SpMatrix(ROWS cat [Universe(ROWS)|V]); // 4 x 5
> Dimension (SpSpace (MAT)); // 3
3
> K := Kernel (MAT); K;
SpSpace of degree 4, dimension 1 over Sp-slope ring of slope 1 given
 by Power series ring in $.1 over pAdicField(7, 30)
Basis given by:
[ (-1 + O(7^30))*u + O(u^21), O(u^21),
  (-7^-1 + O(7^29))*u^2 + O(u^21), 1 + O(7^30) + O(u^21) ]
> KK := Kernel(BasisMatrix(K)); KK;
SpSpace of degree 1, dimension 0 over Sp-slope ring of slope 1 given
 by Power series ring in $.1 over pAdicField(7, 30)
> assert ZeroVector(Ambient(KK)) in KK;

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Example: Modules over Su Rings (ex-9b9567)

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Here is an example over \(S^\nu_u\).

> S<u> := SuRing (pAdicField (7, 20), 5/4); // slope 5/4
> A := SuSpace (S, 3);
> B := SuSpace ([A.1, A.3]); // dimension 2
> SuSpace([B.1]) + SuSpace([7^5/u^4 * B.2]) eq B; // slope 5/4
true
> Basis(sub<A|[A.1+A.3]>)[1] in B;
true
> M := SuMatrix (2, 3, [S | 7^2/u,7,5*7*u, 7,u^3,u^2 ]);
> V := SuSpace(M);
> I := V meet B; assert Dimension(I) eq 1;
> v := Basis(I)[1]; assert v in I;
> WeierstrassTerms(v);
(7 + O(7^21) O($.1^20) $.1^2)
> WeierstrassDegrees(v);
[ 0, -Infinity, 2 ]
> I + B eq B; assert v in B;
true
> J := VerticalJoin (M, M);
> SetSeed(1); // ensure example works
> JJ := (SuMatrixSpace(S,4,4)!RandomSLnZ(4,8,16)) * J;
> assert Dimension(Kernel(JJ)) eq 2;
> assert Dimension(Image(JJ)) eq 2;

Note that the final commands, with the dimension of the kernel/image of the given space, can sometimes fail for various reasons involving precision loss.

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