Block diagonal dominance of matrices revisited: bounds for the norms of inverses and eigenvalue inclusion sets

TL;DR

This paper extends bounds on the inverses of diagonally dominant matrices to block tridiagonal matrices and introduces a Gershgorin-type theorem for tighter eigenvalue inclusion regions.

Contribution

It generalizes classical bounds and the Gershgorin Circle Theorem to block matrices, providing improved spectral analysis tools.

Findings

Derived bounds for inverses of block tridiagonal matrices.

Presented a variant of Gershgorin's theorem for block matrices.

Achieved tighter eigenvalue inclusion regions than previous methods.

Abstract

We generalize the bounds on the inverses of diagonally dominant matrices obtained in [16] from scalar to block tridiagonal matrices. Our derivations are based on a generalization of the classical condition of block diagonal dominance of matrices given by Feingold and Varga in [11]. Based on this generalization, which was recently presented in [3], we also derive a variant of the Gershgorin Circle Theorem for general block matrices which can provide tighter spectral inclusion regions than those obtained by Feingold and Varga.