Theorems · Theorem · nonassociative algebras

RootPairing.Base.exists_cartanMatrix_diagaonal_mul_posDef

∀ {ι : Type u_1} {R : Type u_2} {M : Type u_3} {N : Type u_4} [inst : CommRing R] [inst_1 : AddCommGroup M]
  [inst_2 : Module R M] [inst_3 : AddCommGroup N] [inst_4 : Module R N] {P : RootPairing ι R M N} (b : P.Base)
  [inst_5 : P.IsCrystallographic] [CharZero R] [IsDomain R] [Finite ι] [inst_9 : DecidableEq ι] [P.IsRootSystem],
  ∃ d, (∀ (i : ↥b.support), 0 < d i) ∧ (Matrix.diagonal d * b.cartanMatrix).PosDef

Defined in: Mathlib.LinearAlgebra.RootSystem.CartanMatrix
Cited by: 1 results in Mathlib
Foundations: Depth 128 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module AddCommGroup Module RootPairing.IsCrystallographic CharZero IsDomain Finite DecidableEq RootPairing.IsRootSystem

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Modulestatement and proof · cited by 20,661
CommRingstatement and proof · cited by 17,173
Finsetstatement · cited by 13,712
AddCommGroupstatement and proof · cited by 12,871
Matrixstatement · cited by 4,303
Finitestatement and proof · cited by 3,029
IsDomainstatement and proof · cited by 2,196
CharZerostatement and proof · cited by 932
RootPairingstatement and proof · cited by 710
Matrix.transposeproof · cited by 389
Matrix.diagonalstatement and proof · cited by 314
RootPairing.IsCrystallographicstatement and proof · cited by 160

Cited by1

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RootPairing.Base.det_four_sub_cartanMatrix_ne_zeroproof · cited by 0