apply_apply_same_eq_zero_iff — Mathlib

English

For a symmetric B, the squared CS inequality reduces to the non-squared form via IsSymm.

Русский

Для симметричной B квадрат неравенства CS сводится к не квадратной форме через IsSymm.

LaTeX

$$apply_sq_lt_iff_linearIndependent_of_symm$$

Lean4

theorem apply_apply_same_eq_zero_iff (hs : ∀ x, 0 ≤ B x x) (hB : B.IsSymm) {x : M} : B x x = 0 ↔ x ∈ LinearMap.ker B :=
  by
  rw [LinearMap.mem_ker]
  refine ⟨fun h ↦ ?_, fun h ↦ by simp [h]⟩
  ext y
  have := B.apply_sq_le_of_symm hs hB x y
  simp only [h, zero_mul] at this
  exact pow_eq_zero <| le_antisymm this (sq_nonneg (B x y))

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