add diagonal_with_transform for abstract product spaces

docs/src/quad_forms/basics.md

@@ -115,6 +115,7 @@ gram_matrix(::AbstractSpace{T}, ::Vector{Vector{U}}) where {T, U}
 inner_product(::AbstractSpace, ::Vector, ::Vector)
 orthogonal_basis(::AbstractSpace)
 diagonal(::AbstractSpace)
+diagonal_with_transform(::AbstractSpace)
 restrict_scalars(::AbstractSpace, ::QQField, ::FieldElem)
 ```
 

src/QuadForm/Herm/Spaces.jl

@@ -148,14 +148,22 @@ inner_product(V::HermSpace{S,T,U,W}, v::U, w::U) where {S,T,U,W}= v*gram_matrix(
 #
 ################################################################################
 
-function diagonal(V::HermSpace)
+diagonal(V::HermSpace) = _diagonal(V, false)[1]
+
+diagonal_with_transform(V::HermSpace) = _diagonal(V)
+
+function _diagonal(V::HermSpace, with_transform::Bool = true)
+  E = base_ring(V)
   g = gram_matrix(V)
   k, K = left_kernel(g)
   B = complete_to_basis(K)
   g = B[k+1:end,:]*g*transpose(B[k+1:end,:])
-  D, _ = _gram_schmidt(g, involution(V))
-  D = append!(zeros(base_ring(V),k), diagonal(D))
-  return map(fixed_field(V), D)
+  D, U = _gram_schmidt(g, involution(V))
+  diagE = append!(zeros(base_ring(V),k), diagonal(D))
+  diagK = map(fixed_field(V), diagE)
+  !with_transform && return diagK, E[;]
+  B[k+1:end, :] = U*view(B, k+1:nrows(B), :)
+  return diagK, B
 end
 
 ################################################################################

src/QuadForm/Quad/Spaces.jl

@@ -190,13 +190,21 @@ end
 #
 ################################################################################
 
-function diagonal(V::QuadSpace)
+diagonal(V::QuadSpace) = _diagonal(V, false)[1]
+
+diagonal_with_transform(V::QuadSpace) = _diagonal(V)
+
+function _diagonal(V::QuadSpace, with_transform::Bool = true)
+  E = base_ring(V)
   g = gram_matrix(V)
   k, K = left_kernel(g)
   B = complete_to_basis(K)
   g = B[k+1:end,:]*g*transpose(B[k+1:end,:])
-  D, _ = _gram_schmidt(g, involution(V))
-  return append!(zeros(base_ring(V),k),diagonal(D))
+  D, U = _gram_schmidt(g, involution(V))
+  diag = append!(zeros(base_ring(V), k), diagonal(D))
+  !with_transform && return diag, E[;]
+  B[k+1:end, :] = U*view(B, k+1:nrows(B), :)
+  return diag, B
 end
 
 ################################################################################

src/QuadForm/Spaces.jl

@@ -1,7 +1,7 @@
 export ambient_space, rank, gram_matrix, inner_product, involution, is_hermitian, is_quadratic, is_regular,
        is_local_square, is_isometric, is_rationally_isometric, is_isotropic, is_isotropic_with_vector, quadratic_space,
        hermitian_space, diagonal, invariants, hasse_invariant, witt_invariant, orthogonal_basis, fixed_field,
-       restrict_scalars, orthogonal_complement, orthogonal_projection
+       restrict_scalars, orthogonal_complement, orthogonal_projection, diagonal_with_transform
 
 ################################################################################
 #
@@ -319,6 +319,19 @@ The elements are contained in the fixed field of `V`.
 """
 diagonal(V::AbstractSpace)
 
+@doc raw"""
+    diagonal_with_transform(V::AbstractSpace) -> Vector{FieldElem},
+                                                             MatElem{FieldElem}
+
+Return a vector of elements $a_1,\dotsc,a_n$ such that the space `V` is
+isometric to the diagonal space $\langle a_1,\dotsc,a_n \rangle$. The second
+output is a matrix `U` whose rows span an orthogonal basis of `V` for which the
+Gram matrix is given by the diagonal matrix of the $a_i$'s.
+
+The elements are contained in the fixed field of `V`.
+"""
+diagonal_with_transform(V::AbstractSpace)
+
 ################################################################################
 #
 #  Gram-Schmidt

test/QuadForm/Herm/Spaces.jl

@@ -130,3 +130,12 @@
   @test is_isometric(V1, V2)
 
 end
+
+@testset "diagonal with transform" begin
+  E, b = cyclotomic_field_as_cm_extension(3)
+  K = base_field(E)
+  s = involution(E)
+  Vh = hermitian_space(E, E[1 b 1; s(b) 4 s(b); 1 b 1])
+  diag, U = @inferred diagonal_with_transform(Vh)
+  @test diagonal(map_entries(K, U*gram_matrix(Vh)*map_entries(s, transpose(U)))) == diag
+end

test/QuadForm/Quad/Spaces.jl

@@ -524,3 +524,10 @@ end
   @test !is_isotropic(V, 5)
   @test_throws ArgumentError is_isotropic(V, 4)
 end
+
+@testset "diagonal with transform" begin
+  K, a = cyclotomic_field(8)
+  V = quadratic_space(K, K[a 3 a; 3 3 3; a 3 a])
+  diag, U = @inferred diagonal_with_transform(V)
+  @test diagonal(U*gram_matrix(V)*transpose(U)) == diag
+end