Generalized eigenvalues of max and min matrices

TheoremGeneralized eigenvalues of a MAX matrix to a MIN matrix#●ready to prove

Let \(n > 2\) and let \(0 < s_1 < s_2 < \dots < s_n\) be real numbers. Let \(M\) be the \(n \times n\) matrix with entries \(M_{ij} = \max(s_i, s_j)\), let \(N\) be the \(n \times n\) matrix with entries \(N_{ij} = \min(s_i, s_j)\), and put \(\rho = \sqrt{s_n / s_1}\). Then, as polynomials in \(X\) with real coefficients,

\[\det(M - XN) = (-1)^n \det(N) \, (X - \rho)(X + 1)^{n-2}(X + \rho).\]

The roots of \(\det(M - XN)\) are the generalized eigenvalues of \(M\) to \(N\).

public theorem_wanted det_maxMatrix_sub_X_smul_minMatrix (n : ℕ) (hn : 2 < n)
    (s : Fin n → ℝ) (hs : StrictMono s) (hpos : 0 < s ⟨0, by omega⟩) :
    ((Matrix.of fun i j => max (s i) (s j)).map C -
        (X : ℝ[X]) • (Matrix.of fun i j => min (s i) (s j)).map C).det =
      C ((-1) ^ n * (Matrix.of fun i j => min (s i) (s j)).det) *
        ((X - C (Real.sqrt (s ⟨n - 1, by omega⟩ / s ⟨0, by omega⟩))) *
          (X + 1) ^ (n - 2) *
          (X + C (Real.sqrt (s ⟨n - 1, by omega⟩ / s ⟨0, by omega⟩))))

Sources

• Jorma K. Merikoski, Pentti Haukkanen, Antonio Sasaki, and Timo Tossavainen, On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices, Journal of Integer Sequences 28 (2025), Article 25.7.1, Theorem 1 (jis-2025-25.7.1-thm-1).

TheoremCauchy interlacing for generalized eigenvalues#●ready to prove

Let \(n \ge 1\), let \(A\) be a Hermitian and \(B\) a positive definite complex \((n+1) \times (n+1)\) matrix, and let \(A'\) and \(B'\) be their leading principal \(n \times n\) submatrices (rows and columns \(1, \dots, n\)). Let \(\lambda_1 \ge \dots \ge \lambda_{n+1}\) be real numbers such that the roots of the polynomial \(\det(A - XB)\), counted with multiplicity, are exactly \(\lambda_1, \dots, \lambda_{n+1}\), and let \(\lambda'_1 \ge \dots \ge \lambda'_n\) be real numbers such that the roots of \(\det(A' - XB')\), counted with multiplicity, are exactly \(\lambda'_1, \dots, \lambda'_n\). Then \(\lambda_{k+1} \le \lambda'_k \le \lambda_k\) for every \(k = 1, \dots, n\).

public theorem_wanted generalizedEigenvalue_interlacing (n : ℕ) (hn : 1 ≤ n)
    (A B : Matrix (Fin (n + 1)) (Fin (n + 1)) ℂ) (hA : A.IsHermitian) (hB : B.PosDef)
    (lam : Fin (n + 1) → ℝ) (hlam : Antitone lam)
    (hroots : (A.map C - (X : ℂ[X]) • B.map C).det.roots =
      Finset.univ.val.map fun i => (lam i : ℂ))
    (lam' : Fin n → ℝ) (hlam' : Antitone lam')
    (hroots' : ((A.submatrix Fin.castSucc Fin.castSucc).map C -
        (X : ℂ[X]) • (B.submatrix Fin.castSucc Fin.castSucc).map C).det.roots =
      Finset.univ.val.map fun i => (lam' i : ℂ)) :
    ∀ k : Fin n, lam k.succ ≤ lam' k ∧ lam' k ≤ lam k.castSucc

Sources

• Jorma K. Merikoski, Pentti Haukkanen, Antonio Sasaki, and Timo Tossavainen, On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices, Journal of Integer Sequences 28 (2025), Article 25.7.1, Theorem 6 (jis-2025-25.7.1-thm-6).

2 ready to prove

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