The problem of finding necessary and sufficient conditions for a list of real numbers to be the spectrum of a nonnegative symmetric matrix is called symmetric nonnegative inverse eigenvalue problem (SNIEP).
The authors first present a sufficient condition for an SNIEP in Theorem 3.1. Let \(\Lambda =\{\lambda_1, \lambda_2, \dots, \lambda_{n+1}\}\), \(\mu =\{\mu_1, \mu_2, \dots, \mu_{n}\}\) be two lists of real numbers such that \(\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \dots \geq \mu_{n} \geq \lambda_{n+1}\), with \(\sum_{k=1}^{n+1}\lambda_k-\sum_{k=1}^{n}\mu_k \geq 0\). Let \(P\) be an orthogonal matrix, \(D=\operatorname{diag}\{\mu_1, \mu_2, \dots, \mu_n\}\) such that \(PDP^T \geq 0\), and \(\mathbf{y}=[ y_1 \;y_2 \;\dots \;y_n]^T\) with \(y_i^2 = -f(\mu_i)/g'(\mu_i)\) such that \(P\mathbf{y} \geq 0\), where the functions \(f(t) = \prod_{k=1}^{n+1}(t-\lambda_k)\), \(g(t) = \prod_{k=1}^{n}(t-\mu_k)\). Then, there exists a nonnegative symmetric matrix \(A\) with spectrum \(\Lambda\).
Secondly, a necessary condition for an SNIEP is given in Theorem 3.4. If \(\Lambda =\{\lambda_1, \lambda_2, \dots, \lambda_{n+1}\}\) is realizable for a nonnegative symmetric matrix, then there exist a list of real numbers \(\mu =\{\mu_1, \mu_2, \dots, \mu_{n}\}\) such that \(\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \dots \geq \mu_{n} \geq \lambda_{n+1}\), and an orthogonal matrix \(Q\) and \(\mathbf{b} \in \mathbb{R}^{n\times 1}\) such that \(Q\mathbf{b} \geq 0\) and \(QDQ^T \geq 0\), where \(D=\operatorname{diag}\{\mu_1, \mu_2, \dots, \mu_n\}\). Also it holds that \(\sum_{k=1}^{n+1}\lambda_k-\sum_{k=1}^{n}\mu_k \geq 0\).
These conditions are independent of the existing realizability criteria. To show that, some examples are provided. The authors also show that these two conditions establish a necessary and sufficient condition for the SNIEP when \(n \leq 4\), and conjecture that it holds for \(n \geq 5\).
Finally, a programming algorithm is shown in relation to the above results.