Nilpotent matrices and spectrally arbitrary sign patterns

Abstract

It is shown that all potentially nilpotent full sign patterns are spectrally arbitrary. A related result for sign patterns all of whose zeros lie on the main diagonal is also given. 1. Full Spectrally Arbitrary Patterns. In what follows, Mn denotes the topological vector space of all n × n matrices with real entries and Pn denotes the set of all polynomials with real coefficients of degree n or less. The superdiagonal of an n × n matrix consists of the n − 1 elements that are in the ith row and (i +1 )st column for some i ,1 ≤ i ≤ n − 1. A sign pattern is a matrix with entries in {+, 0, −} .G iven twon×n sign patterns A and B ,w e say thatB is a superpattern of A if bij = aij whenever aij � Note that a sign pattern is always a superpattern of itself. We define the function sign : R →{ +, 0, −} in the obvious way: sign(x )=+i f x> 0, sign(0) = 0, and sign(x )= − if x< 0. Given a real matrix A, sign(A) is the sign pattern with the same dimensions as A whose (i, j)th entry is sign(aij). For every sign pattern A ,w e define its associated sign pattern class to be the inverse image Q(A )= sign −1 (A). A sign pattern is said to be full if none of its entries are zero (8). A sign pattern class