Red blood cell indices

2013-05-14T19:45:45Z

89.234.118.230: /* Mean corpuscular hemoglobin concentration */

In the [[mathematics|mathematical]] field of [[linear algebra]] and [[convex analysis]], the '''numerical range''' or '''field of values''' of a [[complex number|complex]] ''n'' × ''n'' [[square matrix|matrix]] ''A'' is the set

:<math>W(A) = \left\{\frac{\mathbf{x}^*A\mathbf{x}}{\mathbf{x}^*\mathbf{x}} \mid \mathbf{x}\in\mathbb{C}^n,\ x\not=0\right\} </math>

where '''x'''* denotes the [[Hermitian adjoint]] of the [[vector (mathematics)|vector]] '''x'''.

In engineering, numerical ranges are used as a rough estimate of [[eigenvalue]]s of ''A''. Recently, generalizations of numerical range are used to study [[quantum computing]].

A related concept is the '''numerical radius''', which is the largest absolute values of the numbers in the numerical range, i.e.

:<math>r(A) = \sup \{ |\lambda| : \lambda \in W(A) \} = \sup_{\|x\|=1} |\langle Ax, x \rangle|.</math>

''r''(''A'') is a [[norm (mathematics)|norm]].

==Properties==

# The numerical range is the [[range of a function|range]] of the [[Rayleigh quotient]].
# ('''Hausdorff–Toeplitz theorem''') The numerical range is convex and compact.
# <math>W(\alpha A+\beta I)=\alpha W(A)+\{\beta\}</math> for all square matrix ''A'' and complex numbers α and β. Here ''I'' is the [[identity matrix]].
# <math>W(A)</math> is a subset of the closed right half-plane if and only if <math>A+A^*</math> is positive semidefinite.
# The numerical range <math>W(\cdot)</math> is the only function on the set of square matrices that satisfies (2), (3) and (4).
# (Sub-additive) <math>W(A+B)\subseteq W(A)+W(B)</math>.
# <math>W(A)</math> contains all the [[eigenvalue]]s of ''A''.
# The numerical range of a 2×2 matrix is an [[elliptical disk]].
# <math>W(A)</math> is a real line segment [α, β] if and only if ''A'' is a [[Hermitian matrix]] with its smallest and the largest eigenvalues being α and β
# If ''A'' is a [[normal matrix]] then <math>W(A)</math> is the convex hull of its eigenvalues.
# If α is a sharp point on the boundary of <math>W(A)</math>, then α is a normal eigenvalue of ''A''.
# <math>r(\cdot)</math> is a norm on the space of ''n''×''n'' matrices.
# <math>r(A^n) \le r(A)^n</math>

== Generalisations ==
*[[C - numerical range]]
*[[Higher rank numerical range]]
*[[Joint numerical range]]
*[[Product numerical range]]

==See also==
*[[Spectral theory]]
* [[Rayleigh quotient]]

==References==
{{reflist|2}}

;Bibliography
*{{Citation|last1=Choi|first1=M.D.|last2=Dribs|first2=D.W.|last3=Życzkowski|title=Quantum error correcting codes from the compression formalism |journal=Rep. Math. Phys., 58, 2006 | year=2006}}.
*{{Citation|last1=Dirr|first1=G.|last2=Helmkel|first2=U.|last3=Kleinsteuber|first3=M.|last4=Schulte-Herbrüggen|first4=Th.|title=A new type of C-numerical range arising in quantum computing| journal=Proc. Appl. Math. Mech. 6, 711–712 (2006) | year=2006}}.
*{{Citation | last1=Bonsall | first1=F.F. | last2=Duncan | first2=J. | title=Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras | publisher=[[Cambridge University Press]] | isbn=978-0-521-07988-4 | year=1971}}.
*{{Citation | last1=Bonsall | first1=F.F. | last2=Duncan | first2=J. | title=Numerical Ranges II | publisher=[[Cambridge University Press]] | isbn=978-0-521-20227-2 | year=1971}}.
*{{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Topics in Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-46713-1 | year=1991}}.
*{{Citation |last1=Li| first1=C.K.| title=A simple proof of the elliptical range theorem |
journal=Proc. Am. Math. Soc. 124, 1985 | year=1996}}.
*{{Citation |last1=Keeler| first1=Dennis S.|last2=Rodman| first2=Leiba|last3=Spitkovsky|first3=Ilya M.|
title=The numerical range of <math> 3 \times 3 </math> matrices |
journal=Linear Algebra Applications 252, 115 | year=1997}}.
* 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.

{{DEFAULTSORT:Numerical Range}}
[[Category:Matrix theory]]
[[Category:Spectral theory]]
[[Category:Operator theory]]
[[Category:Linear algebra]]

formulasearchengine - User contributions [en]

Red blood cell indices