Theorems · Theorem · linear algebra

Matrix.lt_two_mul_of_mul_diagonal_posDef_of_for_le_of_hasEigen

∀ {ι : Type u_1} {R : Type u_2} [inst : Fintype ι] [inst_1 : DecidableEq ι] [inst_2 : CommRing R]
  [inst_3 : LinearOrder R] [IsStrictOrderedRing R] [inst_5 : StarRing R] [TrivialStar R] (A : Matrix ι ι R) (d : ι → R),
  (Matrix.diagonal d * A).PosDef →
    (∀ (i : ι), 0 < d i) →
      ∀ (μ ρ : R),
        (∀ (i j : ι), A i j ≤ if i = j then μ else 0) → Module.End.HasEigenvalue (Matrix.toLin' A) ρ → ρ < 2 * μ

A spectral bound result for Z-matrices satisfying a positive-definiteness condition. It is important because it applies to Cartan matrices, and shows that all (real) eigenvalues must be strictly less than 4 (a fact which is necessary in the proof of RootPairing.GeckConstruction.instIsIrreducible).

Defined in

Mathlib.LinearAlgebra.Matrix.ZMatrix

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeDecidableEqCommRingLinearOrderIsStrictOrderedRingStarRingTrivialStar