Theorems · Theorem · operator theory

LinearMap.IsIdempotentElem.isSymmetric_iff_isOrtho_range_ker

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {T : E →ₗ[𝕜] E}, IsIdempotentElem T → (T.IsSymmetric ↔ T.range ⟂ T.ker)

An idempotent operator is symmetric if and only if its range is pairwise orthogonal to its kernel.

Defined in

Mathlib.Analysis.InnerProductSpace.Symmetric

Cited by

1 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpace