Geršgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices. II. Some determinantal identities
Joel L. Brenner · Pacific Journal of Mathematics · 1971
A square matrix A = [α^]f has dominant diagonal if Vΐίl an \ > Ri = Σj^i I ttΐj |}.A more complicated type of dominance is the following.Suppose for each i, there is assigned a set I(ϊ) (subset of {1, , n}), i e I(i): Define Ba as the I(i) x I(i) submatrix of A that uses columns I(i) 9 and rows {I(i)\i, j}, i.e., the set obtained from I(i) by replacing the ith row by the jth row.Set ba -det Bij.Then [6^]f is a matrix, the elements of which are determinants of minor matrices of A. In an earlier paper, bounds for det A were derived in case [bij] has dominant diagonal in the special case that {I(i)}% represents a partitioning of the indices into disjoint subsets.In this article the general case is treated; I(i) can be any subset of {1, •••,?&} that contains i.An identity is derived connecting det [6^]f with det A.