dedekind::linear_algebra::detail Namespace Reference

Classes
struct	ComplexToMatrix2x2Hom
struct	DualToMatrix2x2Hom

Typedefs
using	Rat = dedekind::numbers::Rational<dedekind::numbers::default_integer>

Variables
constexpr Complex< Rat >	z_probe {Rat{1L}, Rat{2L}}
constexpr Complex< Rat >	w_probe {Rat{3L}, Rat{-5L}}
constexpr Dual< Rat >	d_probe {Rat{2L}, Rat{3L}}
constexpr Dual< Rat >	e_probe {Rat{-1L}, Rat{5L}}
constexpr Dual< Rat >	eps_q {Rat{0L}, Rat{1L}}
constexpr Matrix2x2V< Rat >	nc_a {Rat{0L}, Rat{1L}, Rat{0L}, Rat{0L}}
constexpr Matrix2x2V< Rat >	nc_b {Rat{0L}, Rat{0L}, Rat{1L}, Rat{0L}}
constexpr Matrix2x2V< Rat >	decomp_probe
constexpr Vec2V< Rat >	tr_vec {Rat{7L}, Rat{-3L}}
constexpr Covec2V< Rat >	tr_covec {Rat{2L}, Rat{5L}}
constexpr Matrix2x2V< Rat >	orth_I = identity_matrix2x2_v<Rat>
constexpr Matrix2x2V< Rat >	orth_R90 {Rat{0L}, Rat{-1L}, Rat{1L}, Rat{0L}}
constexpr Matrix2x2V< Rat >	orth_Rx {Rat{1L}, Rat{0L}, Rat{0L}, Rat{-1L}}
constexpr Matrix2x2V< Rat >	orth_R180 = orth_R90 * orth_R90
constexpr Vec2V< Rat >	col_e1 {Rat{1L}, Rat{0L}}
constexpr Vec2V< Rat >	col_e2 {Rat{0L}, Rat{1L}}

Detailed Description

embeddings__Ring_Homomorphism_Witnesses_over_ℚ

The paper-facing existential proofs that both embeddings are ring homomorphisms — exact at compile time over ℚ = Rational<default_integer> (the saturating-cardinal field, not the signed-long shortcut).

— the :contracts slogan-pack

witnessed on the concrete Matrix2x2V / Vec2V / Covec2V triple.

The nine structural claims, shape-conformance, transpose involution / duality, and the orthogonal group O(2, ℚ) — each pinned by static_assert. If a :contracts concept is tightened in a way the concrete carriers fail, the build breaks here.

Variable Documentation

decomp_probe

Matrix2x2V<Rat> dedekind::linear_algebra::detail::decomp_probe

inlineconstexpr

Initial value:

{Rat{1L}, Rat{2L}, Rat{3L},
                                              Rat{4L}}

nc_a

Matrix2x2V<Rat> dedekind::linear_algebra::detail::nc_a {Rat{0L}, Rat{1L}, Rat{0L}, Rat{0L}}

inlineconstexpr

— algebra concepts on ℚ carriers.

Rather than redefine the vector-space / module content, witness that the value-level tuple and matrix carriers satisfy the canonical operator-shape predicates living one level up: HasFieldOperators (in dedekind.algebra:field), HasRingOperators (in dedekind.algebra:ring), and HasVectorSpaceOperators (in dedekind.algebra:vectorspace). This anchors the new tuple / matrix types to the upstream algebraic hierarchy without redefining its concept surface here.

The_Nine_Matrix_Slogans

orth_I

Matrix2x2V<Rat> dedekind::linear_algebra::detail::orth_I = identity_matrix2x2_v<Rat>

inlineconstexpr

(2,_ℚ) — the multiplicative group of orthogonal matrices.

Concrete rational orthogonal matrices. For each one we check MᵀM = M·Mᵀ = I. Composition of orthogonals is orthogonal (group closure); transpose is the inverse (group inverse law).

tr_vec

Vec2V<Rat> dedekind::linear_algebra::detail::tr_vec {Rat{7L}, Rat{-3L}}

inlineconstexpr

— type-level ill-typing of mismatched shapes.

Transpose_Duality_and_Involution