Bounds on norms of compound matrices and on products of eigenvalues

An upper bound on operator norms of compound matrices is presented and special cases that involve the l l and l norms are investigated The results are then used to obtain bounds on products of the largest or smallest eigenvalues of a matrix The research was started while the rst author visited the Technion in It was continued during a visit of the second author at Universit at Bielefeld and at the University of Wisconsin Madison in and it was completed during a visit of the third author at the Technion in The research was partly supported by SFB Diskrete Strukturen in der Mathematik Bielefeld and by NSF Grant DMS Introduction Let A be a complex matrix and let Ck A be its kth compound It was shown in Formula that the maximal row sum of moduli of elements of Ck A is less than or equal to the product of the k largest rows sums of A and it follows that the product of k largest moduli of eigenvalues of A is bounded above by the product of the k largest row sums of A The case of equality in these inequalities investigated in Theorems I and II The results in and can be viewed as relating the l norm of rows of a matrix to the l norm of its compounds viewed as an operator on rows Working in terms of columns we consider in this paper the relations between other norms of columns and norms of the compounds We begin by proving a general result of the above type which involves a constant k We evaluate this constant in some special cases that involve the l l and l norms Again this leads naturally to upper bounds on the product of the k largest eigenvalues or equivalently lower bounds on the product of the k smallest eigenvalues which involve products of norms of columns and of norms of rows of the matrix As a consequence of our theorems we obtain generalizations of results of and on bounds on norms of the adjoint matrix which is essentially the n compound matrix to kth compound matrices The application of our theorems to the adjoint case sharpens the results in and Upper bounds on norms of compound matrices Let A be a matrix in nn For subsets and of f ng we denote by A j the submatrix of A whose rows are indexed by and whose columns are indexed by in their natural order Let k be a positive integer k n We denote by Ck A the kth compound of the matrix A that is the n k n k matrix whose elements are the minors detA j f ng j j j j k We index Ck A by f ng j j k ordered lexicographically Let be a vector norm on n and for a positive integer k k n let be a submul tiplicative norm on mm where m n k We de ne k maxf Ck B B nn coli B i ng where coli B denotes the ith column of B The following theorem is the main tool from which we derive our results Theorem For an absolute operator norm we have Ck A k max f ng j j k Y