Intensional diagonal and rank-1 outer-product matrix carriers (#372 slice c + e) at any dimension, including ℵ_0. More...

#include <concepts>
#include <cstddef>
#include <functional>
#include <type_traits>
import dedekind.linear_algebra:mat2x2;
import dedekind.sets;
import dedekind.order;
import dedekind.category;
import dedekind.algebra;

Include dependency graph for diagonal.cppm:

Classes
struct	dedekind::linear_algebra::dim_finite< N >
	Type-level witness for a finite dimension `N`. Sibling of `dedekind::sets::ℵ<N>` on the infinite side; together they cover the dimension axis used by `Diagonal` / `OuterProduct` without forcing the `std::variant<>` `Cardinality` value to appear as an NTTP. More...
struct	dedekind::linear_algebra::Diagonal< D, F >
	Diagonal matrix carrier `M:` `D×D → T parametric in` dimension `D` and diagonal rule `F`. More...
struct	dedekind::linear_algebra::identity_rule< T >
	Identity rule: `i` `↦` `T{1}`. IsArrow witness for the constant unit. More...
struct	dedekind::linear_algebra::zero_rule< T >
	Zero rule: `i` `↦` `T{0}`. More...
struct	dedekind::linear_algebra::one_hot< T, D >
	The one-hot / basis vector `e_k` : `j` `↦` `δ(j,k)` `=` [j=k] (`T{1}` on its own index `k`, `T{0}` elsewhere) — a 1-arg `IsArrow`, the `k-th` COLUMN of the identity Identity (the Kronecker δ with one leg fixed; Diagonal 's entry is `δ_ij·F`). More...
struct	dedekind::linear_algebra::diagonal_product_rule< F, G >
	The componentwise rule (F`·G)`(i) `=` `F(i)` `·` `G(i)`. More...
struct	dedekind::linear_algebra::OuterProduct< U, V, Mult >
	Rank-1 matrix `M` `=` `u⊗v with M[i`][j] `=` `u(i)⊗v(j), the` product taken in a semiring. More...
struct	dedekind::linear_algebra::detail_diag::linear_rule_int
struct	dedekind::linear_algebra::detail_diag::const_rule_three
struct	dedekind::linear_algebra::detail_diag::const_rule_six
struct	dedekind::linear_algebra::detail_diag::rule_one_two

Namespaces
namespace	dedekind::linear_algebra::detail_diag

Concepts
concept	dedekind::linear_algebra::IsDimension
	A type-template witness for a dimension cardinality. Today's inhabitants: `dim_finite<N>` and `dedekind::sets::ℵ<N>`.

Typedefs
template<typename D, typename T>
using	dedekind::linear_algebra::Identity = Diagonal<D, identity_rule<T>>
	Identity matrix as `Diagonal<D`, `identity_rule<T>>`.
template<typename D, typename T>
using	dedekind::linear_algebra::DiagonalZero = Diagonal<D, zero_rule<T>>
	Zero diagonal as `Diagonal<D`, `zero_rule<T>>`.

Functions
template<typename D, typename F, typename G> requires IsDimension<D> && dedekind::category::IsArrow<F> && dedekind::category::IsArrow<G>
constexpr auto	dedekind::linear_algebra::operator* (Diagonal< D, F > const &a, Diagonal< D, G > const &b)
	Diagonal-times-diagonal: returns the componentwise-product diagonal at the same dimension `D`.
template<typename F> requires dedekind::category::IsArrow<F> && std::semiregular<typename std::remove_cvref_t<F>::Codomain> && dedekind::algebra::HasRingOperators< typename std::remove_cvref_t<F>::Codomain>
constexpr Matrix2x2V< typename std::remove_cvref_t< F >::Codomain >	dedekind::linear_algebra::to_dense (Diagonal< dim_finite< 2 >, F > const &d)

Variables
constexpr Identity< dim_finite< 2 >, int >	dedekind::linear_algebra::detail_diag::id2 {}
constexpr Identity< dedekind::sets::ℵ_0, int >	dedekind::linear_algebra::detail_diag::id_inf {}
constexpr Diagonal< dedekind::sets::ℵ_0, linear_rule_int >	dedekind::linear_algebra::detail_diag::diag_lin {}
constexpr OuterProduct< linear_rule_int, const_rule_three >	dedekind::linear_algebra::detail_diag::rk1 {}
constexpr OuterProduct< linear_rule_int, const_rule_six >	dedekind::linear_algebra::detail_diag::rk1_scaled {}
constexpr Diagonal< dim_finite< 2 >, rule_one_two >	dedekind::linear_algebra::detail_diag::diag_one_two {}
constexpr auto	dedekind::linear_algebra::detail_diag::diag_one_two_squared = diag_one_two * diag_one_two

Detailed Description

Intensional diagonal and rank-1 outer-product matrix carriers (#372 slice c + e) at any dimension, including ℵ_0.

@partition :diagonal

Copyright: 2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

diagonal__Motivation

Two intensional matrix carriers that bypass dense storage and therefore extend uniformly to infinite dimension ℵ_0:

Diagonal<D, F> — a diagonal matrix M: D×D → T storing the rule F: ℕ → T with M[i][j] = δ_ij · F(i). The canonical instance Identity<D, T> fixes F to the constant i ↦ T{1}.
OuterProduct<U, V> — the rank-1 matrix M = u⊗v with M[i][j] = u(i)·v(j), where u and v are IsArrow factors with size_t domain and a common scalar codomain. Either factor may be infinite-dimensional (e.g. sequences::Path<T> over ℵ_0); the outer product never materialises a dense m× n storage.

Together they discharge the named trigger of #372 ("first infinite-dim tuple/matrix carrier lands") via two complementary intensional shapes: the diagonal subalgebra of End(ℓ²), and the rank-1 image of ℓ² ⊗ ℓ² in the dense matrix algebra.

diagonal__Type_Level_Dimension_Witnesses

dim_finite<N> (defined here) — finite dimension N as an NTTP, is_finite = true. Mirrors std::integral_constant but with the is_finite / is_countable discriminators the cardinality ladder uses.
dedekind::sets::ℵ<N> (existing) — is_finite = false, is_countable = (N==0). Use ℵ_0 (= ℵ<0>) for countable infinity.

The IsDimension concept selects either via the is_finite / is_countable surface, without committing to a std::variant<> NTTP (which the numbers::natural precedent (#402) navigated for runtime values, but type-template parameters keep the carrier-side dispatch simpler at the type level).

Wikipedia: Diagonal matrix, Outer product, Tensor product of vector spaces

Note: "Wir müssen wissen, wir werden wissen." [Trans: "We must know, we will know."] — David Hilbert, Königsberg radio address (8 September 1930); inscribed on his tombstone in Göttingen. Applied here to the rule-based carriers: an intensional matrix is its rule — at any cardinality, every entry is decidable on demand without a dense representation.

Function Documentation

◆ to_dense()

template<typename F>
requires dedekind::category::IsArrow<F> && std::semiregular<typename std::remove_cvref_t<F>::Codomain> && dedekind::algebra::HasRingOperators< typename std::remove_cvref_t<F>::Codomain>

Matrix2x2V< typename std::remove_cvref_t< F >::Codomain > dedekind::linear_algebra::to_dense ( Diagonal< dim_finite< 2 >, F > const & d )

constexprexport

— Diagonal<dim_finite<2>, F>

→ Matrix2x2V<T> at finite n=2 (#372 slice d).

At finite dimension, the intensional Diagonal<dim_finite<N>, F> has a canonical dense companion: the N×N matrix with F(i) on the diagonal and T{} off-diagonal. At N = 2, that companion is Matrix2x2V<T> from :mat2x2. The arrow is one-way: every diagonal carrier projects onto a dense matrix; the reverse direction (recognising rank-≤2 sparsity in a dense matrix) is not universally well-defined and stays out of scope.

This is the bridge between the intensional rank-1 / diagonal carriers from #534 and the existing 2×2 worked instance from #500. The diagonal subalgebra of End(V_2(T)) = M_2(T) lives literally inside Matrix2x2V<T> under this arrow.

Classes

Namespaces

Concepts

Typedefs