Matrices of dual numbers are isomorphic to dual numbers over matrices

Showing this for the more general case of TrivSqZeroExt R M would require an action between Matrix n n R and Matrix n n M, which would risk causing diamonds.

def Matrix.dualNumberEquiv {R n : Type} [CommSemiring R] [Fintype n] [DecidableEq n] : Matrix n n (DualNumber R) ≃ₐ[R] DualNumber (Matrix n n R)

Matrices over dual numbers and dual numbers over matrices are isomorphic.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Matrix.dualNumberEquiv_symm_apply {R n : Type} [CommSemiring R] [Fintype n] [DecidableEq n] (d : DualNumber (Matrix n n R)) : dualNumberEquiv.symm d = of fun (i j : n) => (TrivSqZeroExt.fst d i j, TrivSqZeroExt.snd d i j)

@[simp]

theorem Matrix.dualNumberEquiv_apply {R n : Type} [CommSemiring R] [Fintype n] [DecidableEq n] (A : Matrix n n (DualNumber R)) : dualNumberEquiv A = (of fun (i j : n) => TrivSqZeroExt.fst (A i j), of fun (i j : n) => TrivSqZeroExt.snd (A i j))