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

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

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.

Examples

* For the identity matrix, $scriptstyle\; F(I)\; =\; 1$.

Properties

Let $A,\; B$ be matrices and $sigma(A)$ denote the set of eigenvalues of $A$.
* If $alpha$ is a scalar, then $F(alpha\; A\; )\; =\; alpha\; F(A)$.
* The mapping $xmapsto\; x^ast\; A\; x$ 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 $sigma(A)$ if $A$ is normal.
* $F(A)$ is a subset of the closed right half-plane if and only if $A\; +\; A^*$ is positive semidefinite.

ee also

* Rayleigh quotient

References

* 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.