Theorems · Theorem · linear algebra
Matrix.posSemidef_iff_dotProduct_mulVec
∀ {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.PosSemidef ↔ M.IsHermitian ∧ ∀ (x : n → R), 0 ≤ star x ⬝ᵥ M.mulVec xA finite matrix M : Matrix n n R is positive semidefinite iff it is
Hermitian and xᴴ * M * x is nonnegative for all x.
- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Fintypestatement and proof · cited by 7,736
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Finsuppproof · cited by 5,255
- Finset.sumproof · cited by 5,195
- Matrixstatement and proof · cited by 4,303
- Finset.univproof · cited by 3,473
- Finiteproof · cited by 3,029
- Finset.sum_congrproof · cited by 2,323
- MulZeroClass.mul_zeroproof · cited by 2,091
- StarRingstatement and proof · cited by 1,686
Cited by5
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.of_dotProduct_mulVec_nonnegproof · cited by 5
- Matrix.PosSemidef.dotProduct_mulVec_nonnegproof · cited by 4
- Matrix.PosDef.fromBlocks₁₁proof · cited by 1
- LinearMap.isPosSemidef_iff_posSemidef_toMatrixproof · cited by 0
- Matrix.PosSemidef.hadamardproof · cited by 0