Eigenvalue inequalities and perturbation

Courant–Fischer, Weyl, Ky Fan, Cauchy interlacing, Schur–Horn, Eckart–Young, Bauer–Fike, Peierls–Klein for self-adjoint operators; statements only about finite matrices move to LinearAlgebra.

TheoremSchur's theorem on diagonals of self-adjoint operators#✓fully proved

Let \(E\) be a complex inner product space of finite dimension \(n\), let \(T : E \to E\) be a linear operator that is symmetric, that is, \(\langle T x, y \rangle = \langle x, T y \rangle\) for all \(x, y \in E\), let \(\lambda_1, \dots, \lambda_n\) be the eigenvalues of \(T\) listed with multiplicity, and let \(b_1, \dots, b_n\) be an orthonormal basis of \(E\). Then there is a doubly stochastic real \(n \times n\) matrix \(D\) such that \(\operatorname{Re} \langle b_i, T b_i \rangle = \sum_{j=1}^n D_{ij} \lambda_j\) for every \(i\).

theorem schur_diagonal_eigenvalues_doublyStochastic
    {n : ℕ} (T : E →ₗ[ℂ] E) (hT : T.IsSymmetric)
    (hn : Module.finrank ℂ E = n)
    (b : OrthonormalBasis (Fin n) ℂ E) :
    ∃ D : Matrix (Fin n) (Fin n) ℝ,
      D ∈ doublyStochastic ℝ (Fin n) ∧
        (fun i => RCLike.re (inner (𝕜 := ℂ) (b i) (T (b i))) : Fin n → ℝ) =
          D.mulVec (hT.eigenvalues hn)

Sources

• I. Schur, 1923, Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie; A. Horn, Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix, Amer. J. Math. 76 (1954), 620–630, DOI 10.2307/2372705.

Used by 1

Used byPeierls' inequality

• Bauer–Fike theorem
• Courant–Fischer min–max theorem
• Cauchy interlacing theorem
• Ky Fan eigenvalue inequality
• Peierls and Klein inequalities
• Weyl's eigenvalue inequalities

TheoremEckart–Young theorem in operator norm#✓fully proved

Let \(E\) and \(F\) be finite-dimensional inner product spaces over \(\mathbb{K} = \mathbb{R}\) or \(\mathbb{C}\), let \(T : E \to F\) be a linear map and let \(k \ge 0\) be an integer. The infimum of the operator norms \(\|T - S\|\) over all linear maps \(S : E \to F\) whose range has dimension at most \(k\) equals \(\sigma_k(T)\), where \(\sigma_0(T) \ge \sigma_1(T) \ge \cdots\) are the singular values of \(T\) listed with multiplicity and indexed from \(0\), and \(\sigma_j(T) = 0\) for \(j \ge \dim E\).

theorem eckartYoung_best_rank_approx_operator_norm
    (T : E →ₗ[𝕜] F) (k : ℕ) :
    sInf
        { r : ℝ |
          ∃ S : E →ₗ[𝕜] F,
            Module.finrank 𝕜 S.range ≤ k ∧
              r = ‖(T - S).toContinuousLinearMap‖ } =
      T.singularValues k

Sources

• C. Eckart and G. Young, The Approximation of One Matrix by Another of Lower Rank, Psychometrika 1 (1936), 211-218, DOI 10.1007/BF02288367; unitary-invariant extension L. Mirsky, QJM 11 (1960), DOI 10.1093/qmath/11.1.50.

2 fully proved

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