Geršgorin theorems, regularity theorems, and bounds for determinants of partitioned matrices. II. Some determinantal identities

J. L. Brenner

Vol. 39 (1971), No. 1, 39–50

DOI: 10.2140/pjm.1971.39.39

Abstract

A square matrix A = [a_ij]₁ⁿ has dominant diagonal if ∀i{|a_ii| > R_i = ∑ _J≠i|a_ij|}. A more complicated type of dominance is the following. Suppose for each i, there is assigned a set I(i) (subset of {1,⋯,n}), i ∈ I(i): Define B_ij as the I(i) × I(i) submatrix of A that uses columns I(i), and rows {I(i)∖i,j}, i.e., the set obtained from I(i) by replacing the i-th row by the j-th row. Set b_ij = detB_ij. Then [b_ij]₁ⁿ is a matrix, the elements of which are determinants of minor matrices of A. In an earlier paper, bounds for detA were derived in case [b_ij] has dominant diagonal in the special case that {I(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,⋯,n} that contains i. An identity is derived connecting det[b_ij]₁ⁿ with detA.

Mathematical Subject Classification 2000

Primary: 15A15

Milestones

Received: 10 September 1970

Revised: 13 April 1971

Published: 1 October 1971