Rank Neg eq zero
QuadraticForm.rankNeg_eq_zero
Plain-language statement
For a positive definite quadratic form, the negative dimension (index) is zero. O'Neill states (p. 47) that "ν = 0 if and only if b is positive semidefinite." Since positive definite implies positive semidefinite (Definitions 17 (1) and (2), p. 46), a positive definite form must have index ν = 0.
Exact Lean statement
theorem rankNeg_eq_zero {E : Type*} [AddCommGroup E]
[Module ℝ E] [FiniteDimensional ℝ E] {q : QuadraticForm ℝ E} (hq : q.PosDef) :
q.negDim = 0Formal artifact
Lean source
theorem rankNeg_eq_zero {E : Type*} [AddCommGroup E] [Module ℝ E] [FiniteDimensional ℝ E] {q : QuadraticForm ℝ E} (hq : q.PosDef) : q.negDim = 0 := by haveI : Invertible (2 : ℝ) := inferInstance unfold QuadraticForm.negDim have h_exists := equivalent_signType_weighted_sum_squared q let w := Classical.choose h_exists have h_no_neg : ∀ i, w i ≠ SignType.neg := QuadraticForm.posDef_no_neg_weights hq (Classical.choose_spec h_exists) simpa [Finset.card_eq_zero, Finset.filter_eq_empty_iff] using h_no_neg- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Mathematics/Geometry/Metric/PseudoRiemannian/Defs.lean:144-153
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