# 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. ```magma > 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 ``` 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 $M$. ```magma > F := 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 ``` ### `Example: transformalt (ex-6727c3)` Another way to obtain a matrix with the same effect as $M$ is to use the function `TransformForm`. ```magma > M1 := TransformForm(J1,"symplectic"); > M2 := TransformForm(J2,"symplectic"); > M_alt := M1*M2^-1; > J1 eq M_alt*J2*Transpose(M_alt); true ``` ### `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](../../Algebras/AlgebrasWithInvolution/index-algebras-with-involution.md#chapstaralgebras) 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. ```magma > 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 ``` ### `Example: conjisom (ex-86d774)` The matrix $M$ constructed in Example [Example: transform](#example-ex-5a3a59) can be used to conjugate the isometry group of $J_1$ to the isometry group of $J_2$. ```magma > F := 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 ``` ## 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. ```magma > F := 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 ``` ### `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](../../references.md#cite-wilson2013)]. ### `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. ```magma > F := 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] ``` ### `Example: skewgs (ex-7c8257)` Alternating forms cannot be diagonalised. ```magma > 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 ```