Pseudoblock conditions of diagonal dominance

The paper presents new pseudoblock diagonal-dominance conditions for matrices with fixed block partitioning, which generalize the pointwise $k$th-order diagonal-dominance conditions and pointwise circuit diagonal-dominance conditions. For matrices satisfying pseudoblock conditions, the singularity/nonsingularity problem is considered. In particular, for block $2\times 2$ matrices, certain known results are improved and generalized. Some eigenvalue inclusion regions corresponding to pseudoblock nonsingularity conditions are presented.