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Numerical range

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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.


  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]


See also[edit]


  • 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.