Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of M- and H-Matrices and Their Inverses

Abstract

The quantity q(A∘ A⁻¹), the minimum modulus of the eigenvalues of A∘ A⁻¹, arises naturally in connection with positive diagonal symmetrizability. For an invertible M-matrix A of order n, the classical bounds 2/n≤ q(A∘ A⁻¹)≤ 1 are known. We discuss the sharpness of the lower bound 2/n and investigate the converse of a related result involving the Jacobi iteration matrix. In particular, we show that ρ(J_{A_k})→ 1 does not, in general, imply q(A_k∘ A_k⁻¹)→ 2/n, and identify a class for which this implication holds. We then turn to invertible H-matrices, a broader class that contains invertible M-matrices. We show that A∘ A⁻¹ is an invertible H-matrix whenever A is an invertible H-matrix. In contrast to the M-matrix setting, q(A∘ A⁻¹) can be arbitrarily close to zero. However, replacing A⁻¹ by the inverse of the comparison matrix restores the classical lower bound: we prove that q(A∘ℳ(A)⁻¹)≥ 2/n and obtain further bounds involving the Jacobi iteration matrix of ℳ(A). Finally, for positive diagonally symmetrizable invertible H-matrices, we establish the upper bound q(A∘ A⁻¹)≤1 and, in the irreducible case, characterize when equality occurs.

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BibTeX RIS

BibTeX

@misc{chauhan2026bounds,
  title = {{Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of $\mathbfM$- and $\mathbfH$-Matrices and Their Inverses}},
  author = {Bharat Pratap Chauhan and Samir Mondal and Sushmitha P},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24019},
  eprint = {2609.24019},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24019},
}

RIS

TY  - UNPB
TI  - Bounds on the Minimum Eigenvalue Modulus for Hadamard Products of M- and H-Matrices and Their Inverses
AU  - Bharat Pratap Chauhan
AU  - Samir Mondal
AU  - Sushmitha P
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24019
UR  - https://arxiv.org/abs/2609.24019
AB  - The quantity q(A∘ A^−1), the minimum modulus of the eigenvalues of A∘ A^−1, arises naturally in connection with positive diagonal symmetrizability. For an invertible M-matrix A of order n, the classical bounds 2/n≤ q(A∘ A^−1)≤ 1 are known. We discuss the sharpness of the lower bound 2/n and investigate the converse of a related result involving the Jacobi iteration matrix. In particular, we show that ρ(J_A_k)→ 1 does not, in general, imply q(A_k∘ A_k^−1)→ 2/n, and identify a class for which this implication holds. We then turn to invertible H-matrices, a broader class that contains invertible M-matrices. We show that A∘ A^−1 is an invertible H-matrix whenever A is an invertible H-matrix. In contrast to the M-matrix setting, q(A∘ A^−1) can be arbitrarily close to zero. However, replacing A^−1 by the inverse of the comparison matrix restores the classical lower bound: we prove that q(A∘ℳ(A)^−1)≥ 2/n and obtain further bounds involving the Jacobi iteration matrix of ℳ(A). Finally, for positive diagonally symmetrizable invertible H-matrices, we establish the upper bound q(A∘ A^−1)≤1 and, in the irreducible case, characterize when equality occurs.
ER  -

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