On the independence of principal minors of determinants

Abstract

INTRODUCTION A prinicipal minor, or coaxial minor, of a determinant A is a minor obtained by striking out from A the same rows as columns. There are 2n1 principal minors of a determi-nant of the nth order, the determinant itself being included, but only n2 n +1 of them are independent.t For n = 1, 2, 3 the principal minors are all independent and for n = 4 the relations between them have been quite extensively studied.t For n>4 little has been published either as to which minors constitute an independent set or as to the relations between the minors. It is one of the purposes of this paper to determine several different types of complete sets of independenit principal minors of the general determinant of the nth order and to show how the elements of the determinant may be expressed in terms of the minors of an independent set. If we have a second determinant of the n th order with elements independent of the elements of the first determinant, there is a definite set of determinants obtained by replacing one or more columns of the first determinant by the corresponding column or columns of the second determinant. The set of principal minors of all such determinants is greatly enlarged over the set from the single determinant. It is the second purpose of this paper to determine several types of complete sets of independent principal minors of this enlarged set. There is also determined in this paper a complete set of independent principal minors of the determinants obtained when the above process is extended by adjoining to the original determinant more than one determinant of the nth order with independent elements.