isSymmetric_iff_orthogonal_range — Mathlib

English

For a linear map T that is idempotent, T is symmetric if and only if the range and kernel are orthogonal.

Русский

Для идемпотентного отображения T симметричность эквивалентна ортогональности образа и ядра.

LaTeX

$$T IsSymmetric ↔ (range T) ⟂ (ker T)$$

Lean4

/-- An idempotent operator `T` is symmetric iff `(range T)ᗮ = ker T`. -/
theorem isSymmetric_iff_orthogonal_range {T : E →L[𝕜] E} (h : IsIdempotentElem T) :
    T.IsSymmetric ↔ (LinearMap.range T)ᗮ = LinearMap.ker T :=
  LinearMap.IsIdempotentElem.isSymmetric_iff_orthogonal_range h.toLinearMap

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