Theorems · Theorem · functional analysis

Matrix.toLin_conjTranspose

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F] {m : Type u_5}
  {n : Type u_6} [inst_5 : Fintype m] [inst_6 : DecidableEq m] [inst_7 : Fintype n] [inst_8 : DecidableEq n]
  [inst_9 : FiniteDimensional 𝕜 E] [inst_10 : FiniteDimensional 𝕜 F] (v₁ : OrthonormalBasis n 𝕜 E)
  (v₂ : OrthonormalBasis m 𝕜 F) (A : Matrix m n 𝕜),
  (Matrix.toLin v₂.toBasis v₁.toBasis) A.conjTranspose = LinearMap.adjoint ((Matrix.toLin v₁.toBasis v₂.toBasis) A)

The linear map associated to the conjugate transpose of a matrix corresponding to two orthonormal bases is the adjoint of the linear map associated to the matrix.

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

4 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFintypeDecidableEqFintypeDecidableEqFiniteDimensionalFiniteDimensional