Localization of the zeros of the permanent of a characteristic matrix

Abstract

It is shown that many bounds for the eigenvalues of a complex matrix A are also bounds for the zeros of the permanent of the characteristic matrix of A. If A = (ai3) is an n-square nonnegative matrix, let 4)(A) be the set of all n-square complex matrices B = (bij) for which Ibijl = aij whenever i # j, and letT(A) be the subset of 4>(A) consisting of all B for which jbiil = a for all i. This note is based on the following. THEOREM 1. If A is an n-square nonnegative matrix then (1) det B $ O VB &(A) => per B O VB eUT(A). PROOF. As Camion and Hoffman [4] have shown, if det B $ 0 for every B e T(A), then there exist a permutation matrix P and a nonsingular diagonal matrix D such that PAD is a dominant diagonal matrix. Brenner [1] has shown that the permanent of a dominant diagonal matrix is nonzero. Hence, since PBD is dominant diagonal, per B = (per PBD)(per D)-1 0 0 VB e'T(A). Consideration of the matrix I 1 A= 1 2 1 1 2shows that the converse of (1) is false. If A is an n-square matrix then each zero of per (zI A) is called a p-root of A. Denote the set of all eigenvalues and the set of all p-roots of A by A(A) and 11(A), respectively. Let 6(A) be the diagonal of the matrix A. An admissible mapping a of a set S of n-square complex matrices is a Received by the editors January 5, 1971. AMS 1970 subject class{fications. Primary 15A15, 15A42; Secondary 15A48.