Existence of a Not Necessarily Symmetric Matrix with Given Distinct Eigenvalues and Graph

TL;DR

This paper proves the existence of real matrices with specified eigenvalues and graph structures, including a result that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix.

Contribution

It establishes conditions under which real matrices with prescribed eigenvalues and graph structures exist, extending classical spectral graph theory.

Findings

Existence of real matrices with given eigenvalues and graph G under certain conditions.

Any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix.

The results connect eigenvalue specifications with graph structures in matrix theory.

Abstract

For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph is G. In particular, this implies that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix.