# The diagonalizable nonnegative inverse eigenvalue problem

## TL;DR

This paper proves that the symmetric nonnegative inverse eigenvalue problem differs from the diagonalizable one, and characterizes the minimal parameter for realizability of a specific eigenvalue list by diagonalizable matrices.

## Contribution

It establishes the fundamental difference between SNIEP and DNIEP and provides a precise value for the minimal parameter in a realizability problem, using Jordan Normal Form analysis.

## Key findings

- SNIEP is not equal to DNIEP.
- Minimal t for realizability of (3+t,3-t,-2,-2,-2) by diagonalizable matrices is 1.
- Diagonalizably realizable lists are distinguished from general realizable lists.

## Abstract

In this paper we prove that the SNIEP $\neq$ DNIEP, i.e. the symmetric and diagonalizable nonnegative inverse eigenvalue problems are different. We also show that the minimum $t>0$ for which $(3+t,3-t,-2,-2,-2)$ is realizable by a diagonalizable matrix is $t=1$, and we distinguish diagonalizably realziable lists from general realizable lists using the Jordan Normal Form