Theorems · Theorem · linear algebra
Matrix.mem_range_scalar_of_commute_transvectionStruct
∀ {n : Type u_1} {R : Type u₂} [inst : DecidableEq n] [inst_1 : CommRing R] [inst_2 : Fintype n] {M : Matrix n n R},
(∀ (t : Matrix.TransvectionStruct n R), Commute t.toMatrix M) → M ∈ Set.range ⇑(Matrix.scalar n)M is a scalar matrix if it commutes with every nontrivial transvection (elementary matrix).
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- DecidableEqCommRingFintype
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- RingHomstatement · cited by 10,189
- Fintypestatement and proof · cited by 7,736
- Set.rangestatement · cited by 4,705
- Matrixstatement and proof · cited by 4,303
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Commutestatement and proof · cited by 639
- mul_addproof · cited by 413
- add_mulproof · cited by 363
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.SpecialLinearGroup.scalar_eq_self_of_mem_centerproof · cited by 2
- Matrix.GeneralLinearGroup.mem_center_iff_val_mem_range_scalarproof · cited by 2
- Matrix.mem_range_scalar_iff_commute_transvectionStructproof · cited by 0