Nilpotent matrices and spectrally arbitrary sign patterns
Rajesh Pereira · Electronic Journal of Linear Algebra · 2007
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. Key words. Nilpotent matrices, Potentially nilpotent sign patterns, Spectrally arbitrary sign patterns. AMS subject classifications. 15A18. 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, −}. Giventwon×n sign patterns A and B, we say that B is a superpattern of A if bij = aij whenever aij ̸ = 0. Note that a sign pattern is always a superpattern of itself. We define the function sign: R →{+, 0, −} in the obvious way: sign(x) =+ifx>0, sign(0) = 0, and sign(x) = − if x<0. Given a real matrix A, sign(A) is the sign pattern with the