Theorems · Theorem · linear algebra
dotProduct_star_self_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}, star v ⬝ᵥ v = 0 ↔ v = 0Note that this applies to ℂ via RCLike.toStarOrderedRing.
- Defined in
- Mathlib.LinearAlgebra.Matrix.DotProduct
- Cited by
- 2 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
- star_mul_self_nonnegproof · cited by 17
- Fintype.sum_eq_zero_iff_of_nonnegproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.conjTranspose_mul_self_eq_zeroproof · cited by 1
- Matrix.trace_conjTranspose_mul_self_eq_zero_iffproof · cited by 1