If J is an n x n matrix, the command VectorSpace(K,n,J) creates a vector space of dimension n over K with a bilinear form whose matrix is J.
The default form attached to every vector space in Magma is the symmetric form whose matrix is the identity matrix.
A vector space V is recognised as a polar space if any of the following conditions apply. (There is no check to ensure that the inner product matrix is is non-degenerate.)
Check if the vector space V is a quadratic space or if the Gram matrix of V is a reflexive form.
The type of the polar space V, returned as a string.
> V := VectorSpace(Rationals(),4); > IsPolarSpace(V); true > PolarSpaceType(V); orthogonal space
The symplectic space of dimension n defined by the n x n matrix J. This function checks to ensure that J is alternating.
Returns true if the Involution attribute of the generic vector space W is not assigned and the space carries an alternating form, otherwise false.Note that a quadratic space over a field of characteristic 2 satisfies these conditions and consequently this function will return true for these spaces.
Given a vector space W over a finite field, this intrinsic returns true if the base field has characteristic 2, the Involution attribute is not assigned to the generic space and the form is symmetric but not alternating, otherwise false.
If V and W are symplectic spaces this function returns the orthogonal direct sum V perp W and linear embeddings V to V perp W and W to V perp W.
In order to accommodate hermitian forms it is necessary to assign a field automorphism of order two to the Involution attribute of the vector space.
Thus a unitary space is characterised as a vector space V whose ambient space, Generic(V), has the attribute Involution and whose inner product matrix is either hermitian or skew hermitian.
The n-dimensional unitary space over the base field K of J, where σ is an automorphism of K of order 2 and where J is an n x n matrix which is hermitian or skew-hermitian with respect to σ.
Return true if the Involution attribute of the generic space of W is assigned and the form is either hermitian or skew-hermitian when restricted to W. The second return value is 1 if the form is hermitian and -1 if the form is skew-hermitian.
> K<z> := GF(25); > J, sigma := StandardHermitianForm(5,K); > J; [ 0 0 0 0 1] [ 0 0 0 1 0] [ 0 0 1 0 0] [ 0 1 0 0 0] [ 1 0 0 0 0] > sigma(z); z^5 > V := UnitarySpace(J,sigma); > u := V![1,z,0,z^2,-1]; > DotProduct(u,u); 0 > InnerProduct(u,u); z^20
The transpose of the matrix σ(M), where σ is an automorphism of the base field of the matrix M.
A vector space V with an attached quadratic form is called a quadratic space. The polar form of a quadratic space is the inner product matrix J of the space. If the characteristic of the field is not 2, the value of the quadratic form on a row vector v is (1/2)(v * J * vtr).
The quadratic space of dimension n defined by the quadratic form represented by an n x n matrix Q. The inner product matrix of the space is Q + Qtr.
The quadratic space of dimension n whose quadratic form is given by the quadratic polynomial f in n variables. If the variables are x1, ..., xn, then for i≤j, the (i, j)-th entry of the matrix of the form is the coefficient of xixj in f.
Provided the characteristic of the field is not two, this is the upper triangular matrix which represents the quadratic form whose polar form is the symmetric matrix J.
The matrix which defines the quadratic form of the quadratic space V.
The value Q(v), where Q is the quadratic form attached to the generic space of the parent of the vector v.
The polynomial ∑i, j qijxixj, where Q = (qij) is the matrix of the quadratic form of the quadratic space V.
The polynomial ∑i, j qijxixj, where Q = (qij) is a matrix representing a quadratic form.
> _<x,y,z> := PolynomialRing(Rationals(),3); > f := x^2 + x*y +3*x*z - 2*y*z + y^2 +z^2; > V := QuadraticSpace(f); > PolarSpaceType(V); quadratic space > IsNonsingular(V); true > QuadraticFormMatrix(V); [ 1 1 3] [ 0 1 -2] [ 0 0 1]
The orthogonal direct sum V of the quadratic spaces V and W together with linear embeddings V to V perp W and W to V perp W.
The tensor product of the quadratic spaces V and W. If β and γ are the symmetric bilinear forms on V and W, respectively, and if ξ is the bilinear form on the tensor product, then ξ(v1 tensor w1, v2 tensor w2) = β(v1, v2)γ(w1, w2).
Given totally singular subspaces U and W of the quadratic space V such that Uperp ∩W = 0 this function returns a totally singular subspace X such that V = X direct-sum Uperp and W⊆X.
If V is a vector space over the finite field K and if J is the Gram matrix of V, the discriminant of V is the determinant Δ of J modulo squares. That is, the discriminant is 0 if Δ is a square in K, 1 if it is a non-square. The form J is required to be non-degenerate.
The Arf invariant of the quadratic space V.Currently this is available only for quadratic spaces of even dimension 2m over a finite field F of characteristic 2. In this case there are two possibilities: either the Witt index of the form is m and the Arf invariant is 0, or the Witt index is m - 1 and the Arf invariant is 1.
The Dickson invariant of the isometry f of the quadratic space V is the rank (mod 2) of 1 - f. If the polar form of Q is non-degenerate, the Dickson invariant defines a homomorphism from the orthogonal group O(V) onto the additive group of order 2.
The spinor norm of the isometry f of the quadratic space V. This is the discriminant of the Wall form (Section Wall Forms) of f.
Given complementary totally singular subspaces U and W of a quadratic space and a basis B for U, return a sequence of pairwise orthogonal hyperbolic pairs whose second components form a basis for W.
The reflection determined by a non-singular vector of a quadratic space.
Given a matrix f representing an isometry of the quadratic space V, return a sequence of vectors such that the product of the corresponding orthogonal reflections is f. The empty sequence is returned if f is the identity matrix.
Given a matrix f representing an isometry of the quadratic space V, return a sequence of reflections whose product is f. The empty sequence corresponds to the identity matrix.
Given a quadratic space V defined by a quadratic form Q with polar form β and non-zero vectors u, v ∈V such that u is singular and β(u, v) = 0, the Siegel transformation (also called an Eichler transformation) is the isometry ρu, v defined by
x ρu, v = x + β(x, v)u - β(x, u)v - Q(v)β(x, u) u.
The Siegel transformation defined by a singular vector u and a vector v orthogonal to u. The common parent of u and v must be a quadratic space.
> Q := StandardQuadraticForm(4,3);
> V := QuadraticSpace(Q);
> u := V.1;
> QuadraticNorm(u);
0
> X := { v : v in V | DotProduct(u,v) eq 0 and QuadraticNorm(v) ne 0 };
> #X;
12
> H := sub< GL(V) | [SiegelTransformation(u,v) : v in X]>;
> #H;
9