Field of values

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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.

1 Examples
2 Properties
3 See also
4 References

[edit] Examples

For the identity matrix, F(I) = {1}.

[edit] Properties

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 $\mathbb{C}$ .

[edit] See also

Rayleigh quotient

[edit] 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).

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Categories: Wikipedia articles with topics of unclear importance from November 2006 | Linear algebra | Matrix theory