A Necessary Condition for the Spectrum of Nonnegative Symmetric $ 5 \times 5 $ Matrices

TL;DR

This paper establishes a necessary spectral condition for nonnegative symmetric 5x5 matrices and demonstrates that certain eigenvalue lists satisfying these conditions are realizable, advancing the inverse eigenvalue problem for this matrix size.

Contribution

It provides a new necessary condition for the spectrum of nonnegative symmetric 5x5 matrices and proves its sufficiency under specific inequalities, solving a previously unknown region of the inverse eigenvalue problem.

Findings

Identifies a necessary spectral condition involving eigenvalue sums.

Shows that eigenvalue lists satisfying these conditions are realizable.

Advances the solution to the inverse eigenvalue problem for 5x5 matrices.

Abstract

Let $A$ be a nonnegative symmetric $ 5 \times 5 $ matrix with eigenvalues $ \lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq \lambda_5 $. We show that if $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $ then $ \lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $. McDonald and Neumann showed that $ \lambda_1 + \lambda_3 + \lambda_4 \geq 0 $. Let $ \sigma = \left( \lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5 \right) $ be a list of decreasing real numbers satisfying: 1. $ \sum_{i=1}^{5} \lambda_{i} \geq \frac{1}{2} \lambda_1 $, 2. $ \lambda_3 \leq \sum_{i=1}^{5} \lambda_{i} $, 3. $ \lambda_1 + \lambda_3 + \lambda_4 \geq 0 $, 4. the Perron property, that is $ \lambda_1 = \max_{\lambda \in \sigma} \left| \lambda \right| $. We show that $ \sigma $ is the spectrum of a nonnegative symmetric $ 5 \times 5 $ matrix. Thus, we solve the symmetric nonnegative inverse eigenvalue problem for $ n = 5 $ in a region for which a solution has not been known before.