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
where 
is the conjugate transpose, and 
is the usual Euclidean norm.
The field of values can be used to bound the eigenvalues of sums and products of matrices.
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[edit] Examples
- For the identity matrix, F(I) = 1.
[edit] Properties
Let A,B be matrices and σ(A) denote the set of eigenvalues of A.
- If α is a scalar, then F(αA) = αF(A).
- The mapping
is continuous, and the unit sphere in 
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 
. - The field of values is subadditive:
. - If B is non-singular, then
. As a special case, 
. - 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.
[edit] See also
[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).
- "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.
