Vol. 39, No. 1, 1971

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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

A square matrix A = [aij]1n has dominant diagonal if i{|aii| > Ri = Ji|aij|}. 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 Bij 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 bij = detBij. Then [bij]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 [bij] 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[bij]1n with detA.

Mathematical Subject Classification 2000
Primary: 15A15
Received: 10 September 1970
Revised: 13 April 1971
Published: 1 October 1971
J. L. Brenner