Theorems · Theorem · linear algebra
dotProduct_self_star_eq_zero
∀ {n : Type u_2} {R : Type u_4} [inst : Fintype n] [inst_1 : PartialOrder R] [inst_2 : NonUnitalRing R]
[inst_3 : StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {v : n → R}, v ⬝ᵥ star v = 0 ↔ v = 0Note that this applies to ℂ via RCLike.toStarOrderedRing.
- Defined in
- Mathlib.LinearAlgebra.Matrix.DotProduct
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- PartialOrderstatement and proof · cited by 6,410
- StarRingstatement and proof · cited by 1,686
- Star.starstatement · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- NoZeroDivisorsstatement and proof · cited by 545
- NonUnitalRingstatement and proof · cited by 422
- dotProductstatement · cited by 194
- mul_star_self_nonnegproof · cited by 5
- Fintype.sum_eq_zero_iff_of_nonnegproof · cited by 3
Cited by1
Results whose statement or proof uses this declaration.
- Matrix.self_mul_conjTranspose_eq_zeroproof · cited by 0