A new generalized field of values

Details A new generalized field of values A new generalized field of values

2010-12-17

arxiv.org/abs/1012.3835v1

Mathematics

Numerical Analysis

Scientific paper

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.

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