Theorems · Theorem · linear algebra
Matrix.PosDef.conjTranspose_mul_mul_same
∀ {m : Type u_1} {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
[inst_3 : Fintype n] [inst_4 : Fintype m] {A : Matrix n n R} {B : Matrix n m R},
A.PosDef → Function.Injective B.mulVec → (B.conjTranspose * A * B).PosDef- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
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.starproof · cited by 1,082
- Matrix.mulVecstatement and proof · cited by 267
- Matrix.conjTransposestatement and proof · cited by 202
- dotProductproof · cited by 194
- Matrix.vecMulproof · cited by 148
- Matrix.PosDefstatement and proof · cited by 68
- Matrix.dotProduct_mulVecproof · cited by 19
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.PosDef.mul_mul_conjTranspose_sameproof · cited by 2
- Matrix.IsUnit.posDef_star_left_conjugate_iffproof · cited by 1
- Matrix.PosDef.conjTranspose_mul_selfproof · cited by 0