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Theorems · Theorem · linear algebra

Matrix.PosDef.mul_mul_conjTranspose_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 m n R},
  A.PosDef → (Function.Injective fun v => Matrix.vecMul v B) → (B * A * B.conjTranspose).PosDef
Defined in
Mathlib.LinearAlgebra.Matrix.PosDef
Cited by
2 results in Mathlib
Foundations
Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
RingPartialOrderStarRingFintypeFintype
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Matrix.PosDef.inv · cited by 1PosDef.invMatrix.PosDef.mul_conjTranspose_self · cited by 0PosDef.mul_conjTranspose_…Fintype · cited by 7736FintypeRing · cited by 7463RingPartialOrder · cited by 6410PartialOrderMatrix · cited by 4303MatrixStarRing · cited by 1686StarRingStar.star · cited by 1082Star.starMatrix.mulVec · cited by 267Matrix.mulVecMatrix.conjTranspose · cited by 202Matrix.conjTransposeMatrix.vecMul · cited by 148Matrix.vecMulstar_star · cited by 135star_starMatrix.PosDef · cited by 68Matrix.PosDefMatrix.conjTranspose_conjTranspose · cited by 31Matrix.conjTranspose_conj…star_injective · cited by 8star_injectiveMatrix.PosDef.conjTranspose_mul_mul_same · cited by 3PosDef.conjTranspose_mul_…Matrix.star_vecMul · cited by 1Matrix.star_vecMulPosDef.mul_mul_conjTranspose_…CITED BYCITES

Cites15

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Cited by2

Results whose statement or proof uses this declaration.