Theorems · Theorem · linear algebra
Matrix.PosDef.of_dotProduct_mulVec_pos
∀ {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [inst_3 : Fintype n]
{M : Matrix n n R}, M.IsHermitian → (∀ ⦃x : n → R⦄, x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x) → M.PosDef- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fintypestatement and proof · cited by 7,736
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Matrixstatement and proof · cited by 4,303
- StarRingstatement and proof · cited by 1,686
- Star.starstatement and proof · cited by 1,082
- Matrix.mulVecstatement and proof · cited by 267
- dotProductstatement and proof · cited by 194
- Matrix.IsHermitianstatement and proof · cited by 126
- Matrix.PosDefstatement · cited by 68
- Matrix.posDef_iff_dotProduct_mulVecproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- Matrix.PosDef.conjTranspose_mul_mul_sameproof · cited by 3
- Matrix.PosSemidef.posDef_iff_isUnitproof · cited by 3
- Matrix.PosDef.of_toQuadraticForm'proof · cited by 1
- Matrix.posDef_gram_of_linearIndependentproof · cited by 1