In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.

A transcendental number is one that is not the root of any non-zero polynomial having integer coefficients. It immediately follows that no rational number q is transcendental, since q can be written as a/b where a and b are integers, and thus q is a root of bx − a. Furthermore, the transcendentals are a proper subset of the irrationals, since for example the irrational √ 2 is a root of x − 2. T...