Theorems · Theorem · linear algebra

LinearMap.isNilRegular_def

∀ {R : Type u_1} {L : Type u_2} {M : Type u_3} [inst : CommRing R] [inst_1 : AddCommGroup L] [inst_2 : Module R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] (φ : L →ₗ[R] Module.End R M) [inst_5 : Module.Free R M]
  [inst_6 : Module.Finite R M] [inst_7 : Module.Finite R L] [inst_8 : Module.Free R L] (x : L),
  φ.IsNilRegular x ↔ (LinearMap.charpoly (φ x)).coeff φ.nilRank ≠ 0

Defined in: Mathlib.Algebra.Module.LinearMap.Polynomial
Cited by: 1 results in Mathlib
Foundations: Depth 109 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing AddCommGroup Module AddCommGroup Module Module.Free Module.Finite Module.Finite Module.Free

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Cites13

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
CommRingstatement and proof · cited by 17,173
AddCommGroupstatement and proof · cited by 12,871
LinearMapstatement and proof · cited by 10,215
Polynomial.coeffstatement · cited by 1,045
Module.Finitestatement and proof · cited by 1,032
Module.Endstatement and proof · cited by 774
Module.Freestatement and proof · cited by 597
LinearMap.charpolystatement · cited by 67
LinearMap.nilRankstatement · cited by 10

Cited by1

Results whose statement or proof uses this declaration.

LinearMap.isNilRegular_iff_natTrailingDegree_charpoly_eq_nilRankproof · cited by 2