Theorems · Definition · linear algebra
Module.End.HasEigenvalue
{R : Type v} →
{M : Type w} → [inst : CommRing R] → [inst_1 : AddCommGroup M] → [inst_2 : Module R M] → Module.End R M → R → PropA scalar μ is an eigenvalue for a linear map f if there are nonzero vectors x
such that f x = μ • x. (Def 5.5 of [axler2024]).
- Defined in
- Mathlib.LinearAlgebra.Eigenspace.Basic
- Cited by
- 56 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Module.Endstatement and proof · cited by 774
- Module.End.HasUnifEigenvalueproof · cited by 18
Cited by56
Results whose statement or proof uses this declaration.
- Module.End.HasEigenvalue.exists_hasEigenvectorstatement and proof · cited by 8
- Module.End.hasEigenvalue_of_hasEigenvectorstatement · cited by 8
- Module.End.hasEigenvalue_iffstatement · cited by 5
- Module.End.hasEigenvalue_iff_mem_spectrumstatement · cited by 4
- LinearMap.IsSymmetric.hasEigenvalue_eigenvaluesstatement · cited by 3
- LinearMap.not_hasEigenvalue_zero_tfaestatement and proof · cited by 2
- Module.End.HasEigenvalue.mem_spectrumstatement and proof · cited by 2
- Module.End.hasEigenvalue_of_isRootstatement · cited by 2
- LinearMap.finrank_maxGenEigenspace_zero_eqproof · cited by 2
- LinearMap.hasEigenvalue_zero_tfaestatement and proof · cited by 2
- LinearMap.charpoly_constantCoeff_eq_zero_iffproof · cited by 1
- Module.End.pos_finrank_genEigenspace_of_hasEigenvaluestatement and proof · cited by 1