If β is a bilinear or sesquilinear form on the vector space V, a linear transformation g of V is an isometry if g preserves β; it is a similarity if it preserves β up to a non-zero scalar multiple.
If J is an n x n matrix which represents a bilinear form and if M is a non-singular n x n matrix, then J and MJM^(tr) are said to be congruent and they define isometric polar spaces.
Conversely, given bilinear forms J1 and J2 the following example shows how to use the IsIsometric function to determine whether J1 and J2 are congruent and if so how to find a matrix M such that J1 = MJ2M^(tr).
Returns true if the map f is an isometry from U to V with respect to the attached forms.
Returns true if the map f is an isometry from its domain to its codomain.
Returns true if the matrix g is an isometry of V with respect to the attached form.
Determines whether the polar spaces V and W are isometric; if they are, an isometry is returned (as a map).
> F := GF(5); > V1 := VectorSpace(F,5); > PolarSpaceType(V1); orthogonal space > WittIndex(V1); 2 > J2 := StandardSymmetricForm(5,F); > J2; [0 0 0 0 1] [0 0 0 1 0] [0 0 2 0 0] [0 1 0 0 0] [1 0 0 0 0] > V2 := VectorSpace(F,5,J2); > IsIsometric(V1,V2); false > V3 := VectorSpace(F,5,2*J2); > flag, f := IsIsometric(V1,V3); flag; true > IsIsometry(f); true
> F<x> := GF(25); > J1 := Matrix(F,4,4,[ 0, x^7, x^14, x^13, x^19, 0, x^8, x^5, > x^2, x^20, 0, x^17, x, x^17, x^5, 0 ]); > J2 := Matrix(F,4,4,[ 0, x^17, 2, x^23, x^5, 0, x^15, x^5, > 3, x^3, 0, 4, x^11, x^17, 1, 0 ]); > V1 := SymplecticSpace(J1); > V2 := SymplecticSpace(J2); > flag, f := IsIsometric(V1,V2); assert flag; > f; Mapping from: ModTupFld: V1 to ModTupFld: V2 given by a rule > M := Matrix(F,4,4,[f(V1.i) : i in [1..4]]); > J1 eq M*J2*Transpose(M); true
> M1 := TransformForm(J1,"symplectic"); > M2 := TransformForm(J2,"symplectic"); > M_alt := M1*M2^-1; > J1 eq M_alt*J2*Transpose(M_alt); true
A common complement to the subspaces U and W in the vector space V. (The subspaces must have the same dimension.) This is used by the following function, which implements Witt's theorem.
An extension of the isometry f : U to V to an isometry V to V, where U is a subspace of the polar space V.This is an implementation of Witt's theorem on the extension of an isometry defined on a subspace of a symplectic, unitary or quadratic space. The isometry f must satisfy f(U∩rad(V)) = f(U)∩rad(V).
If the characteristic is two and the form J of V is symmetric, then J must be alternating.
The group of isometries of the polar space V. This includes degenerate polar spaces as well as polar spaces defined by a quadratic form over a field of characteristic two.
Given a reflexive form J, the function IsometryGroup(J) defined in Chapter ALGEBRAS WITH INVOLUTION returns the isometry group of J. More generally, if S is a sequence of reflexive forms, the function IsometryGroup(S) returns the group of isometries of the system.
> F := GF(4); > Q1 := StandardQuadraticForm(4,F : Minus); > Q := DiagonalJoin(Q1,ZeroMatrix(F,2,2)); > V := QuadraticSpace(Q); > G := IsometryGroup(V); > [ IsIsometry(V,g) : g in Generators(G) ]; [ true, true, true, true, true, true, true ] > #G; 96259276800
> F<x> := GF(25); > J1 := Matrix(F,4,4,[ 0, x^7, x^14, x^13, x^19, 0, x^8, x^5, > x^2, x^20, 0, x^17, x, x^17, x^5, 0 ]); > J2 := Matrix(F,4,4,[ 0, x^17, 2, x^23, x^5, 0, x^15, x^5, > 3, x^3, 0, 4, x^11, x^17, 1, 0 ]); > V1 := SymplecticSpace(J1); > V2 := SymplecticSpace(J2); > flag, f := IsIsometric(V1,V2); assert flag; > M := Matrix(F,4,4,[f(V1.i) : i in [1..4]]); > G1 := IsometryGroup(V1); > G2 := IsometryGroup(V2); > M^-1*G1.1*M in G2; true > M^-1*G1.2*M in G2; true
If β is a bilinear or sesquilinear form, a linear transformation f is a similarity with multiplier λ if β(uf, vf) = λβ(u, v) for all u, v.
Returns true if the map f is a similarity from U to V with respect to the attached forms. The second return value is the multiplier.
Returns true if the map f is a similarity from its domain to its codomain. The second return value is the multiplier.
Returns true if the matrix g is a similarity of V with respect to the attached form. The second return value is the multiplier.
Determines whether the polar spaces V and W are similar; if they are, a similarity is returned (as a map).
> F<z> := GF(25); > sigma := hom< F -> F | x :-> x^5 >; > J1 := Matrix(F,4,4,[ > 0, z^3, z^14, z^9, z^15, 2, z^21, z^5, > z^22, z^9, 1, z^7, z^21, z, z^11, 4]); > J2 := Matrix(F,4,4,[ > z^15, z^10, z^17, z^7, z^14, z^15, z^14, z^9, > z, z^10, z^3, z^20, z^23, z^9, z^16, z^21]); > V1 := UnitarySpace(J1,sigma); > V2 := UnitarySpace(J2,sigma); > IsUnitarySpace(V1); true 1 > IsUnitarySpace(V2); true -1 > IsIsometric(V1,V2); false > flag, f := IsSimilar(V1,V2); > flag; true > IsSimilarity(V1,V2,f); true z^3
The group of similarities of the polar space V. This includes degenerate polar spaces as well as polar spaces defined by a quadratic form over a field of characteristic two.
Suppose that β is a bilinear or sesquilinear form on the vector space V and that e1, ..., en is an ordered basis for V. The Gram matrix J of β with respect to this basis is (β(ei, ej)). Changing the basis to e1A, ..., enA where A is an invertible matrix changes the Gram matrix to the congruent matrix AJAσ t, where σ is a field automorphism. (When the form is symmetric or alternating σ is the identity.)
For a positive definite inner product on a real vector space it is possible to choose A so that AJAt is the identity matrix. The algorithm which achieves this is the Gram--Schmidt process.
In general it is possible to choose A so that D = AJAσ t
is almost diagonal; that is, D is a block diagonal matrix where each
block is either a 1 x 1 block or a 2 x 2 block
(matrix(0&α
sα&0)), where s is -1 if the
form is alternating.
The following intrinsic uses the algorithm of Wilson [Wil13].
Given the matrix J of a reflexive form, apply Wilson's Gram-Schmidt algorithm to return a pair of matrices D and A such that AJA' = D, where D is almost diagonal and A' is either the transposed conjugate of A (when J is hermitian) or the transpose of A.
> F<i> := QuadraticField(-1); > B := Matrix(F,4,4,[1,0,3-i,2, 0,3,1+i,1+i, 3+i,1-i,0,2, 2,1-i,2,1]); > B; [ 1 0 -i + 3 2] [ 0 3 i + 1 i + 1] [ i + 3 -i + 1 0 2] [ 2 -i + 1 2 1] > D, A := GramSchmidtPair(B); > D; [ 1 0 0 0] [ 0 3 0 0] [ 0 0 -32/3 0] [ 0 0 0 -5/4] > A; [ 1 0 0 0] [ 0 1 0 0] [ -i - 3 1/3*(i - 1) 1 0] [ 1/8*(-i - 4) 1/8*(i - 2) 1/16*(3*i - 7) 1]
> B := Matrix(Rationals(),4,4,[0,0,3,2, 0,0,1,1, -3,-1,0,2, -2,-1,-2,0]); > B; [ 0 0 3 2] [ 0 0 1 1] [-3 -1 0 2] [-2 -1 -2 0] > D, A := GramSchmidtPair(B); > D; [ 0 1 0 0] [-1 0 0 0] [ 0 0 0 1] [ 0 0 -1 0] > A; [ 1 -2 0 0] [ 2 -6 1 0] [-1 3 0 0] [ 0 0 0 1] > A*B*Transpose(A) eq D; true