Theorems · Theorem · linear algebra

Module.End.eigenvectors_linearIndependent

∀ {R : Type v} {M : Type w} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [IsDomain R]
  [Module.IsTorsionFree R M] (f : Module.End R M) (μs : Set R) (xs : ↑μs → M),
  (∀ (μ : ↑μs), f.HasEigenvector (↑μ) (xs μ)) → LinearIndependent (ι := ↑μs) R xs

Eigenvectors corresponding to distinct eigenvalues of a linear operator are linearly independent. (Lemma 5.11 of [axler2024]) We use the eigenvalues as indexing set to ensure that there is only one eigenvector for each eigenvalue in the image of xs. See Module.End.eigenvectors_linearIndependent' for an indexed variant.

Defined in

Mathlib.LinearAlgebra.Eigenspace.Basic

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleIsDomainModule.IsTorsionFree