posSemidef_iff_eq_conjTranspose_mul_self

English

For nonnegative a,b and p≥1, a^p + b^p ≤ (a+b)^p.

Русский

Для неотрицательных a,b и p≥1 выполняется неравенство a^p + b^p ≤ (a+b)^p.

LaTeX

$$$a,b\ge 0, \; p\ge 1 \Rightarrow a^p + b^p \le (a+b)^p$$$

Lean4

/-- A matrix is positive semidefinite if and only if it has the form `Bᴴ * B` for some `B`. -/
@[deprecated CStarAlgebra.nonneg_iff_eq_star_mul_self (since := "2025-09-22")]
theorem posSemidef_iff_eq_conjTranspose_mul_self {A : Matrix n n 𝕜} : PosSemidef A ↔ ∃ (B : Matrix n n 𝕜), A = Bᴴ * B :=
  by classical exact nonneg_iff_posSemidef (A := A) |>.eq ▸ CStarAlgebra.nonneg_iff_eq_star_mul_self

Похожие утверждения