Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices

Let be a block irreducible matrix with nonsingular diagonal blocks, be a positive vector, and let Under these assumptions, necessary and sufficient conditions for A to be singular are obtained based on a block generalization of Wielandt’s lemma. The pointwise case (N=n) of irreducible matrices with nonstrict generalized diagonal dominance is treated separately. For an irreducible matrix A, conditions necessary and sufficient for a boundary point of the union of the Gerschgorin’s circles and of the union of the ovals of Cassini to be an eigenvalue of A are derived.