A note on the symmetric nonnegative inverse eigenvalue problem (Q2905959)

Let \(\Lambda=\{\lambda_1,\dots\lambda_n\}\), \(\lambda_1\geq\dots\geq\lambda_n\), be a (multi)set of real numbers which is the spectrum of an~\(n\times n\) symmetric (entrywise) nonnegative matrix. The symmetric nonnegative inverse eigenvalue problem is to characterize~\(\Lambda\). If \(n\geq 5\), this problem is open; only necessary conditions and sufficient conditions are known.NEWLINENEWLINEThe first-named author, jointly with \textit{O.~Rojo}, \textit{J.~Moro} and \textit{A.~Borobia} [Electron. J. Linear Algebra~16, 1--18 (2007; Zbl 1155.15010)] proved the following: For \(t\leq n\), let \(\omega_1,\dots,\omega_t\) be real numbers satisfying \(0\leq\omega_1,\dots,\omega_t\leq\lambda_1\). Suppose that i)~there exists a partition \(\Lambda=\Lambda_1\cup\cdots\cup\Lambda_t\) with \(\Lambda_k=\{\lambda_{k1},\dots,\lambda_{kp_k}\}\), \(\lambda_{11}=\lambda_1\), \(\lambda_{k1}\geq 0\), \(\lambda_{k1}\geq\dots\geq\lambda_{kp_k}\), \(k=1,\dots,t\), such that, for each~\(k\), the set \(\{\omega_k,\lambda_{k2},\dots,\lambda_{kp_k}\}\) is the spectrum of a symmetric nonnegative \(p_k\times p_k\) matrix; ii)~there exists a symmetric nonnegative \(t\times t\) matrix with eigenvalues \(\lambda_{11},\dots,\lambda_{t1}\) and diagonal entries \(\omega_1,\dots,\omega_t\). Then \(\Lambda\) is the spectrum of a symmetric nonnegative \(n\times n\) matrix.NEWLINENEWLINEThe present authors show that this criterion can be applied to obtain many well-known sufficient conditions for~\(\Lambda\) and to find the corresponding matrix.