Theorems · Theorem · functional analysis

LinearMap.toMatrix_adjoint

∀ {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) (f : E →ₗ[𝕜] F),
  (LinearMap.toMatrix v₂.toBasis v₁.toBasis) (LinearMap.adjoint f) =
    ((LinearMap.toMatrix v₁.toBasis v₂.toBasis) f).conjTranspose

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

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

2 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFintypeDecidableEqFintypeDecidableEqFiniteDimensionalFiniteDimensional