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Mathlib.RingTheory.Finiteness.Nilpotent

Nilpotent maps on finite modules #

theorem Module.End.isNilpotent_iff_of_finite {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [Module.Finite R M] {f : End R M} :
IsNilpotent f ∀ (m : M), ∃ (n : ), (f ^ n) m = 0
@[simp]
theorem Matrix.isNilpotent_iff {R : Type u_1} [CommSemiring R] {ι : Type u_3} [DecidableEq ι] [Fintype ι] {A : Matrix ι ι R} :
IsNilpotent A ∀ (v : ιR), ∃ (n : ), (A ^ n).mulVec v = 0
theorem Matrix.isNilpotent_iff_forall_row {R : Type u_1} [CommSemiring R] {ι : Type u_3} [DecidableEq ι] [Fintype ι] {A : Matrix ι ι R} :
IsNilpotent A ∀ (i : ι), ∃ (n : ), (A ^ n).row i = 0
theorem Matrix.isNilpotent_iff_forall_col {R : Type u_1} [CommSemiring R] {ι : Type u_3} [DecidableEq ι] [Fintype ι] {A : Matrix ι ι R} :
IsNilpotent A ∀ (i : ι), ∃ (n : ), (A ^ n).col i = 0