If βis 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 βup 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
Returns true if the map f is a similarity from U to V with respect to the attached forms.
Returns true if the map f is a similarity from its domain to its codomain.
Returns true if the matrix g is a similarity of V with respect to the attached form.
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.[Next][Prev] [Right] [Left] [Up] [Index] [Root]