dedekind::linear_algebra::IsInvertibleMatrixOverField Concept Reference

A matrix over a field that carries ring-like structure plus an .inverse() operation. More...

Concept definition

template<typename M, typename F>
concept dedekind::linear_algebra::IsInvertibleMatrixOverField = 
    IsMatrixOverFieldRingLike<M, F> && HasMultiplicativeInverse<M>

Detailed Description

A matrix over a field that carries ring-like structure plus an .inverse() operation.

Honest taxonomy (spelled out because the user-level slogan "matrix over a field is sometimes a field if invertible" is deliberately loose):

• Mₙ(F) for n ≥ 2: ring, not division ring, not field.
• GLₙ(F): multiplicative group, not a ring (no additive closure — sum of invertibles can be singular).
• M₁(F) ≅ F: the only collapse where ring = division ring = field all coincide.
• Orthogonal O(n, F): multiplicative group (see IsOrthogonalMatrix).
• Quaternions ℍ: a non-commutative division ring, but not a matrix type in this hierarchy.

This concept therefore captures an honest structural subset: "has ring-like closure and exposes an inverse". It does not claim division- ring status for the carrier.