Theorems · Theorem · linear algebra

hasEigenvalue_toLin_diagonal_iff

∀ {R : Type u_1} {n : Type u_2} {M : Type u_3} [inst : DecidableEq n] [inst_1 : Fintype n] [inst_2 : CommRing R]
  [Nontrivial R] [inst_4 : AddCommGroup M] [inst_5 : Module R M] (d : n → R) {μ : R} [IsDomain R]
  [Module.IsTorsionFree R M] (b : Module.Basis n R M),
  Module.End.HasEigenvalue ((Matrix.toLin b b) (Matrix.diagonal d)) μ ↔ ∃ i, d i = μ

Eigenvalues of a diagonal linear operator are the diagonal entries.

Defined in

Mathlib.LinearAlgebra.Eigenspace.Matrix

Cited by

1 results in Mathlib

Foundations

Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqFintypeCommRingNontrivialAddCommGroupModuleIsDomainModule.IsTorsionFree