Theorems · Theorem · linear algebra
Module.Basis.toMatrix_unitsSMul
∀ {ι : Type u_1} {R₂ : Type u_7} {M₂ : Type u_8} [inst : CommRing R₂] [inst_1 : AddCommGroup M₂] [inst_2 : Module R₂ M₂]
[inst_3 : DecidableEq ι] (e : Module.Basis ι R₂ M₂) (w : ι → R₂ˣ),
e.toMatrix ⇑(e.unitsSMul w) = Matrix.diagonal (Units.val ∘ w)The basis constructed by unitsSMul has vectors given by a diagonal matrix.
- Defined in
- Mathlib.LinearAlgebra.Matrix.Basis
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Matrixstatement · cited by 4,303
- mul_oneproof · cited by 3,885
- Unitsstatement and proof · cited by 2,804
- Units.valstatement and proof · cited by 1,966
- Module.Basisstatement and proof · cited by 1,477
- Finsupp.singleproof · cited by 943
- map_smulproof · cited by 566
- Matrix.extproof · cited by 540
Cited by2
Results whose statement or proof uses this declaration.
- Module.Basis.det_unitsSMul_selfproof · cited by 2
- Module.Basis.toMatrix_isUnitSMulproof · cited by 0