The von Neumann trace inequality and the Hoffman–Wielandt inequality

Let T and S be symmetric endomorphisms of an n-dimensional inner product space over ℝ or ℂ, with eigenvalues λ₀ ≥ ⋯ ≥ λₙ₋₁ and μ₀ ≥ ⋯ ≥ μₙ₋₁ listed in decreasing order.

The diagonal dₖ = Re⟪S eₖ, eₖ⟫ of S in an orthonormal basis (eₖ) is the image of the spectrum μ under the doubly stochastic Schur weight matrix of (eₖ) and an eigenbasis of S. Doubly stochastic averaging cannot increase a sorted pairing, so for every decreasing weight sequence c,

∑ₖ cₖ dₖ ≤ ∑ₖ cₖ μₖ

(LinearMap.IsSymmetric.sum_mul_re_inner_apply_self_le_sum_mul_eigenvalues, in the Schur–Horn file).

Taking (eₖ) to be an eigenbasis of T and c = λ gives the von Neumann trace inequality Re tr (T S) ≤ ∑ᵢ λᵢ μᵢ.

Expanding ∑ₖ ‖(T - S) eₖ‖² = ∑ᵢ λᵢ² - 2 Re tr (T S) + ∑ᵢ μᵢ² then gives the Hoffman–Wielandt inequality: the squared Euclidean distance between the sorted spectra is at most the squared Frobenius distance between the operators,

∑ᵢ (λᵢ - μᵢ)² ≤ ∑ₖ ‖T eₖ - S eₖ‖²,

for every orthonormal basis (eₖ).

Main results

• LinearMap.IsSymmetric.sum_eigenvalues_mul_re_inner_apply_self_le: for e an eigenbasis of T, ∑ₖ λₖ Re⟪S eₖ, eₖ⟫ ≤ ∑ₖ λₖ μₖ.
• LinearMap.IsSymmetric.re_trace_comp_le_sum_eigenvalues_mul_eigenvalues: the von Neumann trace inequality Re tr (T S) ≤ ∑ᵢ λᵢ μᵢ.
• LinearMap.IsSymmetric.sum_sq_eigenvalues_sub_le_sum_sq_norm_apply: the Hoffman–Wielandt inequality ∑ᵢ (λᵢ - μᵢ)² ≤ ∑ₖ ‖T eₖ - S eₖ‖².

References

• J. von Neumann, Some matrix-inequalities and metrization of matric-space, Tomsk Univ. Rev. 1 (1937), 286–300.
• A. J. Hoffman, H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37–39.
• R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer (1997), §VI.4.

theorem LinearMap.IsSymmetric.sum_eigenvalues_mul_re_inner_apply_self_le {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {T S : E →ₗ[𝕜] E} {n : ℕ} (hT : T.IsSymmetric) (hS : S.IsSymmetric) (hn : Module.finrank 𝕜 E = n) : ∑ k : Fin n, hT.eigenvalues hn k * RCLike.re (inner 𝕜 (S ((hT.eigenvectorBasis hn) k)) ((hT.eigenvectorBasis hn) k)) ≤ ∑ i : Fin n, hT.eigenvalues hn i * hS.eigenvalues hn i

The von Neumann trace core. For symmetric T and S, pairing the eigenvalues of T with the diagonal of S in an eigenbasis of T gives at most the sorted pairing ∑ᵢ λᵢ(T) λᵢ(S) of the two spectra.

theorem LinearMap.IsSymmetric.re_trace_comp_le_sum_eigenvalues_mul_eigenvalues {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {T S : E →ₗ[𝕜] E} {n : ℕ} (hT : T.IsSymmetric) (hS : S.IsSymmetric) (hn : Module.finrank 𝕜 E = n) : RCLike.re ((trace 𝕜 E) (T ∘ₗ S)) ≤ ∑ i : Fin n, hT.eigenvalues hn i * hS.eigenvalues hn i

The von Neumann trace inequality. For symmetric T and S with eigenvalues listed in decreasing order, Re tr (T S) ≤ ∑ᵢ λᵢ(T) λᵢ(S).

theorem LinearMap.IsSymmetric.sum_sq_eigenvalues_sub_le_sum_sq_norm_apply {𝕜 : Type u_1} {E : Type u_2} {ι : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [Fintype ι] {T S : E →ₗ[𝕜] E} {n : ℕ} (hT : T.IsSymmetric) (hS : S.IsSymmetric) (hn : Module.finrank 𝕜 E = n) (e : OrthonormalBasis ι 𝕜 E) : ∑ i : Fin n, (hT.eigenvalues hn i - hS.eigenvalues hn i) ^ 2 ≤ ∑ k : ι, ‖T (e k) - S (e k)‖ ^ 2

The Hoffman–Wielandt inequality. For symmetric T and S with eigenvalues listed in decreasing order and every orthonormal basis e, the squared Euclidean distance between the two spectra is at most the squared Frobenius distance ∑ₖ ‖T eₖ - S eₖ‖² between the operators.