Theorems · Theorem · functional analysis
OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary
∀ {ι : Type u_1} {𝕜 : Type u_3} [inst : RCLike 𝕜] {E : Type u_4} [inst_1 : NormedAddCommGroup E]
[inst_2 : InnerProductSpace 𝕜 E] [inst_3 : Fintype ι] [inst_4 : DecidableEq ι] (a b : OrthonormalBasis ι 𝕜 E),
a.toBasis.toMatrix ⇑b ∈ Matrix.unitaryGroup ι 𝕜The change-of-basis matrix between two orthonormal bases a, b is a unitary matrix.
- Defined in
- Mathlib.Analysis.InnerProductSpace.PiL2
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- NormedAddCommGroupstatement and proof · cited by 15,752
- Fintypestatement and proof · cited by 7,736
- Matrixstatement · cited by 4,303
- InnerProductSpacestatement and proof · cited by 3,523
- Submonoidstatement · cited by 3,086
- RCLikestatement and proof · cited by 2,829
- OrthonormalBasisstatement and proof · cited by 188
- OrthonormalBasis.toBasisstatement · cited by 102
- Module.Basis.toMatrixstatement · cited by 70
- Matrix.unitaryGroupstatement · cited by 37
- Matrix.mem_unitaryGroup_iff'proof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- OrthonormalBasis.det_to_matrix_orthonormalBasis_realproof · cited by 2
- OrthonormalBasis.toMatrix_orthonormalBasis_mem_orthogonalproof · cited by 0
- OrthonormalBasis.det_to_matrix_orthonormalBasisproof · cited by 0