[Paper Review] Inverses of symmetric, diagonally dominant positive matrices and applications

Christopher J. Hillar, Shaowei Lin+1 more|arXiv (Cornell University)|Mar 29, 2012

Graph theory and applications15 references6 citations

TL;DR

This paper establishes tight bounds for the ∞-norm of the inverse of symmetric, diagonally dominant positive matrices, independent of diagonal dominance values Δᵢ(J), and proves a new lower-bound form of Hadamard’s inequality for determinants. The key contribution is a sharp ∞-norm bound of ‖J⁻¹‖∞ ≤ (3n−4)/(2ℓ(n−2)(n−1)) for matrices with entries bounded below by ℓ > 0, with equality iff J = ℓS, where S is the diagonally balanced matrix with off-diagonals 1. The result enables improved numerical stability and consistency in maximum likelihood estimation for maximum entropy graph models.

ABSTRACT

We prove tight bounds for the $\infty$-norm of the inverse of symmetric, diagonally dominant positive matrices. We also prove a new lower-bound form of Hadamard's inequality for the determinant of diagonally dominant positive matrices and an improved upper bound for diagonally balanced positive matrices. Applications of our results include numerical stability for linear systems, bounds on inverses of differentiable functions, and consistency of the maximum likelihood equations for maximum entropy graph distributions.

Motivation & Objective

• To derive sharp, explicit bounds on the ∞-norm of the inverse of symmetric, diagonally dominant positive matrices when diagonal dominance values Δᵢ(J) = 0.
• To establish a new lower-bound form of Hadamard’s inequality for determinants of diagonally dominant positive matrices.
• To prove improved upper bounds for diagonally balanced matrices, independent of the maximum matrix entry.
• To apply these results to numerical stability in linear systems and consistency of maximum likelihood estimators in maximum entropy graph distributions.

Proposed method

• Derive a tight ∞-norm bound for J⁻¹ using the structure of the diagonally balanced matrix S = (n−2)Iₙ + 1ₙ1ₙᵀ, showing that the bound is maximized uniquely at J = ℓS.
• Use matrix majorization and Loewner ordering: if J ≥ ℓS, then J⁻¹ ⪯ (ℓS)⁻¹, and apply unitarily invariant norm inequalities.
• Apply perturbation analysis and reverse Cauchy-Schwarz to bound quadratic forms involving submatrices of J, leading to bounds on determinant ratios.
• Use block matrix determinant identities and eigenvalue analysis to evaluate determinant ratios for structured matrices.
• Establish a recursive bound on the determinant ratio via Schur complements and submatrix norms.
• Prove tightness of bounds by constructing extremal matrices achieving equality, such as J = ℓS and block-structured matrices with controlled entries.

Experimental results

Research questions

• RQ1 What is the tightest possible upper bound on ‖J⁻¹‖∞ for symmetric, diagonally dominant positive matrices when Δᵢ(J) = 0 for all i?
• RQ2 Can a new lower-bound form of Hadamard’s inequality be derived for diagonally dominant positive matrices, independent of matrix entries?
• RQ3 How does the determinant ratio det(J)/∏Jᵢᵢ behave under bounded off-diagonal and diagonal entries, especially in the diagonally balanced case?
• RQ4 Can the ∞-norm bound for J⁻¹ be made independent of the maximum matrix entry, even when Δᵢ(J) = 0?
• RQ5 Is there a combinatorial formula for the entries of (S + tP)⁻¹ for signless Laplacians P, which would aid in proving conjectured bounds?

Key findings

• The ∞-norm of the inverse of any symmetric, diagonally dominant positive matrix with entries bounded below by ℓ > 0 is bounded above by (3n−4)/(2ℓ(n−2)(n−1)), with equality if and only if J = ℓS.
• This bound is strictly tighter than standard spectral norm estimates, which scale as O(1/√n), while the new bound scales as O(1/n).
• For diagonally balanced matrices, the determinant ratio satisfies det(J)/∏Jᵢᵢ ≤ exp(−ℓ²/(4m²)) ≤ exp(−1/4) when m ≥ ℓ, with equality approached in the limit of large m/ℓ.
• The determinant ratio for the matrix S = (n−2)Iₙ + 1ₙ1ₙᵀ satisfies det(S)/∏Sᵢᵢ → 2/e as n → ∞.
• The bound is tight: for any ℓ > 0 and n ≥ 3, equality in the ∞-norm bound is achieved precisely when J = ℓS.
• The paper constructs explicit counterexamples showing that the ∞-norm bound fails to hold without symmetry, and that the norm can diverge under perturbations even in the balanced case.