Block diagonal dominance of matrices revisited: bounds for the norms of inverses and eigenvalue inclusion sets
This work provides theoretical extensions for matrix analysis, benefiting researchers in numerical linear algebra and related fields, but is incremental in nature.
The paper generalizes bounds on inverses of diagonally dominant matrices to block tridiagonal matrices and derives a variant of the Gershgorin Circle Theorem for block matrices, yielding tighter spectral inclusion regions than previous methods.
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.