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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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Chapter 1

The field of values

1.0    Introduction

Like the spectrum (or set of eigenvalues) σ(·), the field of values F(·) is a set of complex numbers naturally associated with a given n-by-n matrix A:

image

The spectrum of a matrix is a discrete point set; while the field of values can be a continuum, it is always a compact convex set. Like the spectrum, the field of values is a set that can be used to learn something about the matrix, and it can often give information that the spectrum alone cannot give. The eigenvalues of Hermitian and normal matrices have especially pleasant properties, and the field of values captures certain aspects of this nice ...

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