dedekind::linear_algebra::DirectSum< A, B > Struct Template Reference

Direct sum of two invertibles — the trivial compositional preserver. More...

Public Types
using	first_factor = A
using	second_factor = B
using	inverse_type
	Inverse: block-diagonal of the factor inverses.

Public Member Functions
constexpr auto	inverse () const
template<typename A2, typename B2>
constexpr auto	operator* (DirectSum< A2, B2 >) const
	Composition: (A ⊕ B) · (A' ⊕ B') = (A·A') ⊕ (B·B').
template<typename A2, typename B2>
constexpr bool	operator== (DirectSum< A2, B2 >) const

Detailed Description

template<typename A, typename B>
struct dedekind::linear_algebra::DirectSum< A, B >

Direct sum of two invertibles — the trivial compositional preserver.

(A ⊕ B) is the block-diagonal operator

[[ A,  0 ],
 [ 0,  B ]]

with (A ⊕ B)^{-1} = A^{-1} ⊕ B^{-1} and (A ⊕ B) · (A' ⊕ B') = (A·A') ⊕ (B·B').

This is Tier 1 of #366: invertibility of a block-diagonal sum is a consequence of invertibility of the blocks, with no Schur-complement recurrence needed. Cheapest move on the block-composition ladder, and the one that builds arbitrarily-large well-behaved matrices from 2×2 bases "for free" by iteration.

The factors A and B are expected to be types that participate in composition (operator* returning a value of the same structural shape) and expose inverse() returning an inverse_type; Invertible2x2<T, …> is the canonical instance today.

Member Typedef Documentation

inverse_type

template<typename A, typename B>

using dedekind::linear_algebra::DirectSum< A, B >::inverse_type

Initial value:

 
      DirectSum<typename A::inverse_type, typename B::inverse_type>

Inverse: block-diagonal of the factor inverses.

The closed form (A ⊕ B)^{-1} = A^{-1} ⊕ B^{-1} is exact whenever each factor's inverse is exact — here, inherited from Invertible2x2 over Rational<Z>.

Member Function Documentation

operator*()

template<typename A, typename B>

template<typename A2, typename B2>

auto dedekind::linear_algebra::DirectSum< A, B >::operator* ( DirectSum< A2, B2 > ) const

inlineconstexpr

Composition: (A ⊕ B) · (A' ⊕ B') = (A·A') ⊕ (B·B').

The two blocks compose independently; the output is again a DirectSum.

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The documentation for this struct was generated from the following file:

• src/main/modules/dedekind/linear_algebra/mat2x2.cppm