Right Right To Matrix symm expand tmul
Fermion.rightRightToMatrix_symm_expand_tmul
Plain-language statement
Expanding rightRightToMatrix in terms of the standard basis.
Exact Lean statement
lemma rightRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) :
rightRightToMatrix.symm M = ∑ i, ∑ j, M i j •
(RightHandedWeyl.basis i ⊗ₜ[ℂ] RightHandedWeyl.basis j)Formal artifact
Lean source
lemma rightRightToMatrix_symm_expand_tmul (M : Matrix (Fin 2) (Fin 2) ℂ) : rightRightToMatrix.symm M = ∑ i, ∑ j, M i j • (RightHandedWeyl.basis i ⊗ₜ[ℂ] RightHandedWeyl.basis j) := by simp only [rightRightToMatrix, LinearEquiv.trans_symm, LinearEquiv.trans_apply, Basis.repr_symm_apply] rw [Finsupp.linearCombination_apply_of_mem_supported ℂ (s := Finset.univ)] · rw [Fintype.sum_prod_type] refine Finset.sum_congr rfl (fun i _ => Finset.sum_congr rfl (fun j _ => ?_)) exact congrArg _ (Basis.tensorProduct_apply RightHandedWeyl.basis RightHandedWeyl.basis i j) · simp- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Fermions/Weyl/Two.lean:119-128
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