Various 3x3 Nonnegative Matrices with Prescribed Eigenvalues and Diagonal Entries

TL;DR

This paper investigates the inverse eigenvalue problem for 3x3 nonnegative matrices with specified eigenvalues and diagonal entries, providing exact ranges and covering various matrix classes including symmetric and stochastic matrices.

Contribution

It offers new results on the inverse eigenvalue problem for 3x3 matrices with prescribed eigenvalues and diagonals, including explicit ranges for diagonal entries across different matrix types.

Findings

Exact ranges for diagonal entries of 3x3 nonnegative matrices

Unified treatment of various matrix classes

Explicit solutions for prescribed eigenvalues and diagonals

Abstract

In this paper, we answer the various forms of nonnegative inverse eigenvalue problems with prescribed diagonal entries for order three: real or complex general matrices, symmetric stochastic matrices, and real or complex doubly stochastic matrices. We include the known cases, the symmetric matrices and real or complex stochastic matrices, to compare the other results and for completeness. In addition, for a given list of eigenvalues, we compute the exact range for the largest value of the diagonal entries of the various nonnegative matrices.