A new generalized field of values

TL;DR

This paper introduces a new generalized field of values for matrices, which is always convex and coincides with the standard field for normal matrices, providing a new perspective on eigenvalue characterization.

Contribution

It proposes a novel generalized field of values based on a Hermitian positive definite matrix linking left and right eigenvectors, extending classical eigenvalue analysis.

Findings

The generalized field of values is always convex and equals the convex hull of eigenvalues.

It coincides with the standard field of values for normal matrices.

Eigenvalues of matrices with real spectrum can be characterized via a generalized Rayleigh Quotient.

Abstract

Given a right eigenvector $x$ and a left eigenvector $y$ associated with the same eigenvalue of a matrix $A$, there is a Hermitian positive definite matrix $H$ for which $y=Hx$. The matrix $H$ defines an inner product and consequently also a field of values. The new generalized field of values is always the convex hull of the eigenvalues of $A$. Moreover, it is equal to the standard field of values when $A$ is normal and is a particular case of the field of values associated with non-standard inner products proposed by Givens. As a consequence, in the same way as with Hermitian matrices, the eigenvalues of non-Hermitian matrices with real spectrum can be characterized in terms of extrema of a corresponding generalized Rayleigh Quotient.