Theorems · Theorem · linear algebra
Matrix.PosDef.smul
∀ {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] {α : Type u_5}
[inst_3 : CommSemiring α] [inst_4 : PartialOrder α] [inst_5 : StarRing α] [StarOrderedRing α] [inst_7 : Algebra α R]
[StarModule α R] [PosSMulStrictMono α R] {x : Matrix n n R}, x.PosDef → ∀ {a : α}, 0 < a → (a • x).PosDef- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Algebrastatement and proof · cited by 11,388
- CommSemiringstatement and proof · cited by 10,911
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Finsuppproof · cited by 5,255
- Matrixstatement and proof · cited by 4,303
- LT.lt.leproof · cited by 2,189
- StarRingstatement and proof · cited by 1,686
- Star.starproof · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- StarModulestatement and proof · cited by 570
- Finsupp.sumproof · cited by 481
Cited by1
Results whose statement or proof uses this declaration.
- RootPairing.Base.exists_cartanMatrix_mul_diagaonal_posDefproof · cited by 1