The minimum number of nonzeros in a spectrally arbitrary ray pattern

A ray pattern AA is spectrally arbitrary if given any monic polynomial r(x)r(x) of order n   with complex coefficients, there exists B∈Qr(A)B∈Qr(A) such that the characteristic polynomial of B   is r(x)r(x). In [1], the authors first presented that a family of n×nn×n irreducible ray patterns with exactly 3n   nonzeros is spectrally arbitrary. Then they proved that every n×nn×n irreducible spectrally arbitrary ray pattern has at least 3n−13n−1 nonzeros. So as pointed in [1], the minimum number of nonzeros in an n×nn×n irreducible spectrally arbitrary ray pattern is either 3n   or 3n−13n−1.In this paper, we provide an irreducible spectrally arbitrary ray pattern of order n   with 3n−13n−1 nonzeros. So the minimum number of nonzeros in an n×nn×n irreducible spectrally arbitrary ray pattern is 3n−13n−1.