Isometries and Similarities

If \(\beta\) is a bilinear or sesquilinear form on the vector space \(V\), a linear transformation \(g\) of \(V\) is an isometry if \(g\) preserves \(\beta\); it is a similarity if it preserves \(\beta\) up to a non-zero scalar multiple.

Isometries

IsIsometry(U, V, f): ModTupFld, ModTupFld, Map -> BoolElt

Returns true if the map \(f\) is an isometry from \(U\) to \(V\) with respect to the attached forms.

IsIsometry(f): Map -> BoolElt

Returns true if the map \(f\) is an isometry from its domain to its codomain.

IsIsometry(V, g): ModTupFld, Mtrx -> BoolElt

Returns true if the matrix \(g\) is an isometry of \(V\) with respect to the attached form.

IsIsometric(V, W): ModTupFld, ModTupFld -> BoolElt, Map

Determines whether the polar spaces \(V\) and \(W\) are isometric; if they are, an isometry is returned (as a map).

Example: isometric (ex-9909e6)

A vector space is always equipped with a bilinear form and the default value is the identity matrix. However the “standard” symmetric form is \(Q+Q^{\hbox{tr}}\), where \(Q\) is the standard quadratic form. In the following example the polar spaces defined by these forms are similar but not isometric.

> 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

Run in calculatorCopy input

If \(J\) is an \(n\times n\) matrix which represents a bilinear form and if \(M\) is a non-singular \(n\times n\) matrix, then \(J\) and \(MJM^{\hbox{tr}}\) are said to be congruent and they define isometric polar spaces.

Conversely, given bilinear forms \(J_1\) and \(J_2\) the following example shows how to use the IsIsometric function to determine whether \(J_1\) and \(J_2\) are congruent and if so how to find a matrix \(M\) such that \(J_1 = MJ_2M^{\hbox{tr}}\).

Example: transform (ex-5a3a59)

Begin with alternating forms \(J_1\) and \(J_2\) over \({\bf F}_{25}\), construct the corresponding symplectic spaces and then use the isometry to define the matrix \(M\).

> 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

Run in calculatorCopy input

Example: transformalt (ex-6727c3)

Another way to obtain a matrix with the same effect as \(M\) is to use the function TransformForm.

> M1 := TransformForm(J1,"symplectic");
> M2 := TransformForm(J2,"symplectic");
> M_alt := M1*M2^-1;
> J1 eq M_alt*J2*Transpose(M_alt);
true

Run in calculatorCopy input

CommonComplement(V, U, W): ModTupFld, ModTupFld, ModTupFld -> ModTupFld

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.

ExtendIsometry(V, U, f): ModTupFld, ModTupFld, Map -> Map

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 \(f(U\cap \hbox{rad}(V)) = f(U)\cap \hbox{rad}(V)\).

If the characteristic is two and the form \(J\) of \(V\) is symmetric, then \(J\) must be alternating.

IsometryGroup(V): ModTupFld -> GrpMat

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.

Example: isometrygroup (ex-833a77)

We give an example of an isometry group of a degenerate quadratic space over a field of characteristic 2.

> 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

Run in calculatorCopy input

Example: conjisom (ex-86d774)

The matrix \(M\) constructed in Example Example: transform can be used to conjugate the isometry group of \(J_1\) to the isometry group of \(J_2\).

> 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

Run in calculatorCopy input

Similarities

If \(\beta\) is a bilinear or sesquilinear form, a linear transformation \(f\) is a similarity with multiplier \(\lambda\) if \(\beta(uf,vf) = \lambda\beta(u,v)\) for all \(u,v\).

IsSimilarity(U, V, f): ModTupFld, ModTupFld, Map -> BoolElt, FldElt

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.

IsSimilarity(f): Map -> BoolElt, FldElt

Returns true if the map \(f\) is a similarity from its domain to its codomain. The second return value is the multiplier.

IsSimilarity(V, g): ModTupFld, Mtrx -> BoolElt, FldElt

Returns true if the matrix \(g\) is a similarity of \(V\) with respect to the attached form. The second return value is the multiplier.

IsSimilar(V, W): ModTupFld, ModTupFld -> BoolElt, Map

Determines whether the polar spaces \(V\) and \(W\) are similar; if they are, a similarity is returned (as a map).

Example: simherm (ex-86e68e)

An example of two unitary spaces, the first defined by an hermitian form, the second by a skew-hermitian form. The spaces are similar but not isometric.

> 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

Run in calculatorCopy input

SimilarityGroup(V): ModTupFld -> GrpMat

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.

Gram-Schmidt Normalisation

Suppose that \(\beta\) is a bilinear or sesquilinear form on the vector space \(V\) and that \(e_1\), …, \(e_n\) is an ordered basis for \(V\). The Gram matrix \(J\) of \(\beta\) with respect to this basis is \((\beta(e_i,e_j))\). Changing the basis to \(e_1A\), …, \(e_nA\) where \(A\) is an invertible matrix changes the Gram matrix to the congruent matrix \(AJA^{\sigma t}\), where \(\sigma\) is a field automorphism. (When the form is symmetric or alternating \(\sigma\) is the identity.)

For a positive definite inner product on a real vector space it is possible to choose \(A\) so that \(AJA^t\) is the identity matrix. The algorithm which achieves this is the Gram–Schmidt process.

In general it is possible to choose \(A\) so that \(D = AJA^{\sigma t}\) is almost diagonal; that is, \(D\) is a block diagonal matrix where each block is either a \(1\times 1\) block or a \(2\times 2\) block \(\left(\begin{matrix}0&\alpha\\ s\alpha&0\end{matrix}\right)\), where \(s\) is \(-1\) if the form is alternating.

The following intrinsic uses the algorithm of Wilson [Wilson, 2013].

GramSchmidtPair(J): AlgMatElt -> AlgMatElt, AlgMatElt

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 \(A\).

Example: hermitiangs (ex-97668f)

For the case of an hermitian matrix over the Gaussian rationals the algorithm is able to diagonalise the form.

> 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]

Run in calculatorCopy input

Example: skewgs (ex-7c8257)

Alternating forms cannot be diagonalised.

> 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

Run in calculatorCopy input