Geršgorin Theorems, Regularity Theorems, and Bounds for Determinants of Partitioned Matrices
Joel L. Brenner · SIAM Journal on Applied Mathematics · 1970
For partitioned matrices, inequalities among the absolute values of certain minor determinants are known to guarantee nonsingularity, and thus to provide root-location theorems. First a new root-location theorem is given. Second, a lemma of Loewy is refined, so that bounds can be obtained for the determinant of a matrix that satisfies the inequality conditions above. Several applications appear. The refinement of Loewy’s lemma consists in the establishing of some determinantal identities; the elements of the determinant are themselves certain minors of an arbitrary $r \times 2r$ (or else an $n \times n$) matrix.