A new generalized field of values
Provides a new theoretical tool for analyzing non-Hermitian matrices, but the contribution is incremental as it builds on existing concepts of field of values and non-standard inner products.
The paper introduces a generalized field of values for non-Hermitian matrices, defined via an inner product constructed from left and right eigenvectors. This new field always equals the convex hull of the eigenvalues, generalizes the standard field of values for normal matrices, and enables a Rayleigh quotient characterization for matrices with real spectrum.
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.