Theorems · Theorem · linear algebra
Module.End.HasEigenvalue.mem_spectrum
∀ {R : Type v} {M : Type w} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {f : Module.End R M}
{μ : R}, f.HasEigenvalue μ → μ ∈ spectrum R f- Defined in
- Mathlib.LinearAlgebra.Eigenspace.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 78 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- 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
- spectrumstatement · cited by 510
- Module.End.HasEigenvaluestatement and proof · cited by 56
- Module.End.HasUnifEigenvalue.mem_spectrumproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.IsHermitian.eigenvalues_mem_spectrum_realproof · cited by 2
- IsCompactOperator.hasEigenvalue_iff_mem_spectrumproof · cited by 1