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