Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex matrix A is the set

where denotes the conjugate transpose of the vector . The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors).

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.

Properties

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  1. The numerical range is the range of the Rayleigh quotient.
  2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
  3. for all square matrix and complex numbers and . Here is the identity matrix.
  4. is a subset of the closed right half-plane if and only if is positive semidefinite.
  5. The numerical range is the only function on the set of square matrices that satisfies (2), (3) and (4).
  6. (Sub-additive) , where the sum on the right-hand side denotes a sumset.
  7. contains all the eigenvalues of .
  8. The numerical range of a matrix is a filled ellipse.
  9. is a real line segment if and only if is a Hermitian matrix with its smallest and the largest eigenvalues being and .
  10. If is a normal matrix then is the convex hull of its eigenvalues.
  11. If is a sharp point on the boundary of , then is a normal eigenvalue of .
  12. is a norm on the space of matrices.
  13. , where denotes the operator norm.[1][2][3][4]

Generalisations

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See also

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Bibliography

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  • Choi, M.D.; Kribs, D.W.; Życzkowski (2006), "Quantum error correcting codes from the compression formalism", Rep. Math. Phys., 58 (1): 77–91, arXiv:quant-ph/0511101, Bibcode:2006RpMP...58...77C, doi:10.1016/S0034-4877(06)80041-8, S2CID 119427312.
  • 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, doi:10.1002/pamm.200610336.
  • 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, Chapter 1, ISBN 978-0-521-46713-1.
  • Horn, Roger A.; Johnson, Charles R. (1990), Matrix Analysis, Cambridge University Press, Ch. 5.7, ex. 21, ISBN 0-521-30586-1
  • Li, C.K. (1996), "A simple proof of the elliptical range theorem", Proc. Am. Math. Soc., 124 (7): 1985, doi:10.1090/S0002-9939-96-03307-2.
  • Keeler, Dennis S.; Rodman, Leiba; Spitkovsky, Ilya M. (1997), "The numerical range of 3 × 3 matrices", Linear Algebra and Its Applications, 252 (1–3): 115, doi:10.1016/0024-3795(95)00674-5.
  • "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.

References

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