English
The product of the previous preorder yields a partial order on matrices; antisymmetry is ensured by the trace criterion for PosSemidef.
Русский
Произведение предыдущего пред-порядка образует частичный порядок на матрицах; антисимметричность обеспечивается критерием по trace через PosSemidef.
LaTeX
$$$ instPartialOrder : PartialOrder (Matrix n n 𝕜) $$$
Lean4
/-- The partial order on matrices given by `A ≤ B := (B - A).PosSemidef`. -/
abbrev instPartialOrder : PartialOrder (Matrix n n 𝕜) where
le_antisymm A B h₁
h₂ := by
rw [← sub_eq_zero, ← h₂.trace_eq_zero_iff]
have := neg_nonneg.mp <| trace_neg (A - B) ▸ neg_sub A B ▸ h₁.trace_nonneg
exact le_antisymm this h₂.trace_nonneg
