Symmetric Nonnegative Inverse Eigenvalue Problem

The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) asks: given a multiset of $n$ real numbers $\sigma = \{\lambda_1, \lambda_2, \dots, \lambda_n\} \subset \mathbb{R}$, does there exist an entrywise nonnegative, symmetric matrix $A \in \mathbb{R}^{n \times n}$ whose spectrum is $\sigma$?

By the Spectral Theorem for real symmetric matrices, any realization $A$ is orthogonally diagonalizable:

This means the entrywise nonnegativity condition $a_{ij} \ge 0$ translates into a system of quadratic inequalities on the rows of an orthogonal matrix $Q = (q_{ik})$:

Because the rows of $Q$ must form an orthonormal basis of $\mathbb{R}^n$ ($\sum_k q_{ik}^2 = 1$ and $\sum_k q_{ik} q_{jk} = 0$ for $i \ne j$), the eigenvectors cannot tilt freely. This rigid orthogonality makes the SNIEP strictly harder to satisfy than general real realizations when dimension $n \ge 5$.

For $n \le 4$, the Symmetric and Real NIEP are identical: any real spectrum realizable non-symmetrically is also realizable symmetrically. However, in 1996, Charles Johnson, Thomas Laffey, and Raphael Loewy proved that for all $n \ge 5$, the SNIEP is strictly smaller than the RNIEP.

When candidate spectra are parameterized by power sum coordinates $(s_1, s_2, \dots, s_k)$, the set of realizable symmetric spectra forms a convex polyhedron called the SNIEP Trace Polytope.

Because $\operatorname{SNIEP}(n) \subsetneq \operatorname{RNIEP}(n)$ for $n \ge 5$, the SNIEP trace polytope sits strictly inside the RNIEP trace polytope, giving researchers an interactive geometric window into the exact boundary where symmetry fails.