In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex \( n\times n \) matrix A is the set
\( W(A)=\left\{{\frac {{\mathbf {x}}^{*}A{\mathbf {x}}}{{\mathbf {x}}^{*}{\mathbf {x}}}}\mid {\mathbf {x}}\in {\mathbb {C}}^{n},\ x\not =0\right\} \)
where \( \mathbf{x}^* \) denotes the conjugate transpose of the vector \( \mathbf{x}^*. \)
In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of the numerical range are used to study quantum computing.
A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e.
\( r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{{\|x\|=1}}|\langle Ax,x\rangle |. \)
Properties
The numerical range is the range of the Rayleigh quotient.
(Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
\( W(\alpha A+\beta I)=\alpha W(A)+\{\beta \} \) for all square matrix A and complex numbers \( \alpha \) and \( \beta \) . Here I is the identity matrix.
W(A) is a subset of the closed right half-plane if and only if \( A+A^{*} \) is positive semidefinite.
The numerical range \( W(\cdot ) \) is the only function on the set of square matrices that satisfies (2), (3) and (4).
(Sub-additive) \( W(A+B)\subseteq W(A)+W(B) \), where the sum on the right-hand side denotes a sumset.
W(A) contains all the eigenvalues of A {\displaystyle A} A.
The numerical range of a \( 2\times 2 \) matrix is a filled ellipse.
W ( A ) {\displaystyle W(A)} W(A) is a real line segment \( {\displaystyle [\alpha ,\beta ]} \) if and only if A is a Hermitian matrix with its smallest and the largest eigenvalues being \( \alpha \) and \( \beta \) .
If A is a normal matrix then W(A) is the convex hull of its eigenvalues.
If α is a sharp point on the boundary of W(A), then \( \alpha \) is a normal eigenvalue of A.
\( r(\cdot ) \) is a norm on the space of \( n\times n \) matrices.
\( {\displaystyle r(A)\leq \|A\|\leq 2r(A)} \), where \( {\displaystyle \|\cdot \|} \) denotes the operator norm.
\( r(A^{n})\leq r(A)^{n} \)
Generalisations
C-numerical range
Higher-rank numerical range
Joint numerical range
Product numerical range
Polynomial numerical hull
See also
Spectral theory
Rayleigh quotient
Workshop on Numerical Ranges and Numerical Radii
Bibliography
Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58: 77, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8.
Dirr, G.; Helmkel, U.; Kleinsteuber, M.; Schulte-Herbrüggen, Th. (2006), "A new type of C-numerical range arising in quantum computing", Proc. Appl. Math. Mech., 6: 711–712.
Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, Cambridge University Press, ISBN 978-0-521-07988-4.
Bonsall, F.F.; Duncan, J. (1971), Numerical Ranges II, Cambridge University Press, ISBN 978-0-521-20227-5.
Horn, Roger A.; Johnson, Charles R. (1991), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1.
Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc., 124: 1985.
Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications, 252: 115.
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.
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