TABLE OF CONTENTS The idea of a principal submatrix of as square matrix was introduced in Section 5.14 where it was pointed out that a principal submatrix of A is any submatrix of A whose principal diagonal is part of the principal diagonal of A.
The expansion of the characteristic equation in Section 7.3 shows how the coefficients p( ?) involve sums of eigenvalues, sums of eigenvalues taken two at a time, sums of eigenvalues taken three at a time on up to product of all of the eigenvalues. This leads to the idea that the characteristic polynomial of A can be formed from a consideration of the principal minors of A that are, of course, determinants of the principal submatrices of A.
The prescription for this is given by
The coefficient that is associated with the k th term in the characteristic polynomial, p( ?), of the order n th order matrix of A is ( ?1) k times the sum of the principal minors of order n ? k of A. Moreover, the coefficient of ? n is ( ?1) n and the constant term is equal to det A.
The matrix
has three first order principal minors given by the elements along the principal diagonal
three second order principal minors
and the principal minor which is the determinant of A,...
