Field of values

In matrix theory, the field of values associated with a matrix is the image of the unit sphere under the quadratic form induced by the matrix.

More precisely, suppose A is a square matrix with complex entries. The field of values for A is the set

$F(A) = \{ x^\ast A x : \Vert x \Vert = 1, x\in \mathbb{C}^n \},$

where $x^\ast$ is the conjugate transpose, and $\Vert \cdot \Vert$ is the usual Euclidean norm.

The field of values can be used to bound the eigenvalues of sums and products of matrices.

[edit] Examples

Let $A, B$ be matrices and $σ(A)$ denote the set of eigenvalues of $A$ .

If $α$ is a scalar, then $F (α A) = α F (A)$ .
The mapping $x\mapsto x^\ast A x$ is continuous, and the unit sphere in $\mathbb{C}^n$ is compact. Therefore the field of values is always compact. By the Heine–Borel theorem, it follows that F(A) is closed and bounded in $\mathbb{C}$ .
The field of values is subadditive: $F(A+B) \subseteq F(A) + F(B)$ .
If $B$ is non-singular, then $\sigma(B^{-1}A) \subseteq F(A) / F(B)$ . As a special case, $\sigma(A) \subseteq F(A)$ .
$F (A)$ is convex. It is the convex hull of $σ(A)$ if $A$ is normal.
$F (A)$ is a subset of the closed right half-plane if and only if $A + A *$ is positive semidefinite.

Roger A. Horn and Charles R. Johnson, Topics in Matrix Analysis, Chapter 1, Cambridge University Press, 1991. ISBN 0-521-30587-X (hardback), ISBN 0-521-46713-6 (paperback).
"Functional Characterizations of the Field of Values and the Convex Hull of the Spectrum", Charles R. Johnson, Proceedings of the American Mathematical Society, 61(2):201-204, Dec 1976.