Theorems · Theorem · linear algebra

Matrix.isSymmetric_toLin_iff

∀ {𝕜 : Type u_1} {n : Type u_3} {A : Matrix n n 𝕜} [inst : RCLike 𝕜] [inst_1 : Fintype n] [inst_2 : DecidableEq n]
  {E : Type u_4} [inst_3 : NormedAddCommGroup E] [inst_4 : InnerProductSpace 𝕜 E] (b : OrthonormalBasis n 𝕜 E),
  ((Matrix.toLin b.toBasis b.toBasis) A).IsSymmetric ↔ A.IsHermitian

A matrix is Hermitian iff the corresponding linear map with an orthonormal basis is symmetric.

Defined in

Mathlib.Analysis.Matrix.Hermitian

Cited by

2 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeFintypeDecidableEqNormedAddCommGroupInnerProductSpace