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Properties of numerical range

Properties of numerical range $W(A)$ of a matrix $A$ of dimension $d$ [1], [2]:

  1. $W(A)$ is a compact subset of $\mathbb{C}$,
  2. $W(A)$ is convex set (Hausdorff-Toeplitz theorem) [3], [4],
  3. $W(A)$ contains the spectrum of $A$,
  4. if $AA^\dagger=A^\dagger A$ ($A$ is a normal matrix) then $W(A)$ is convex hull of spectrum of $A$,
  5. if $A=A^\dagger$ then $W(A)=[\lambda_1, \lambda_N]$ forms an interval in the real axis,
  6. if $N=2$ then $W(A)$ forms an elliptic disk with eigenvalues $\lambda_1$ and $\lambda_2$ as focal points and the minor axis $d = \sqrt{\tr(AA^\dagger) - |\lambda_1|^2- |\lambda_2|^2}$ [5].
  7. if $N=3$ we distinguish four cases (classification by Keeler, Rodman, Spitkovsky) [6]:
    1. $W(A)$ is a compact set of an 'ovular' shape containing three eigenvalues (the generic case),
    2. $W(A)$ is a compact set with one flat part (e.g. convex hull of a cardioid),
    3. $W(A)$ is a compact set with two flat parts (e.g. convex hull of an ellipse and a point outside it),
    4. $W(A)$ if a triangle with eigenvalues in its corners (in the case of normal $A$)

More interesting properties you can find in [7], [8], [9], [10], [11], [12], [13], [14], [15].


An example application of numerical range can be found in [16]:


For a list of examples, see examples of numerical range.


1. R. Horn, C. Johnson, 1994. Topics in matrix analysis. Cambridge university press.
2. K. E. Gustafson, D. K. M. Rao, 1997. Numerical range: The Field of Values of Linear Operators and Matrices. Springer.
3. O. Toeplitz, 1918. Das algebraische Analogon zu einem Satze von Fejer. Mathematische Zeitschrift, 2, Springer, pp.187–197.
4. F. Hausdorff, 1919. Der Wertevorrat einer Bilinearform. Mathematische Zeitschrift, 3, Springer, pp.314–316.
5. F. D Murnaghan, 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, National Academy of Sciences, pp.246.
6. D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. Linear Algebra and its Applications, 252, pp.115 - 139.
7. Mao-Ting Chien, Hiroshi Nakazato, Jie Meng, 2019. The diameter and width of numerical ranges. Linear Algebra and its Applications, 582, Elsevier, pp.76–98.
8. Jean-Christophe Bourin, Antoine Mhanna, 2017. Positive block matrices and numerical ranges. Comptes Rendus Mathematique, 355, Elsevier, pp.1077–1081.
9. Nam-Kiu Tsin, 1983. Diameter and minimal width of the numerical range. Linear and multilinear algebra, 14, Taylor and Francis, pp.179–185.
10. Titas Geryba, Ilya M Spitkovsky, 2020. On the numerical range of some block matrices with scalar diagonal blocks. Linear and Multilinear Algebra, Taylor and Francis, pp.1–14.
11. Chi-Kwong Li, Yiu-Tung Poon, 2019. Numerical Range Inclusion, Dilation, and Operator Systems. arXiv preprint arXiv:1911.01221.
12. Mahsa Fatehi, Asma Negahdari, 2019. Numerical range of weighted composition operators which contain zero. arXiv preprint arXiv:1901.07736.
13. Jaedeok Kim, Youngmi Kim, 2018. Jordan Plane and Numerical Range of Operators Involving Two Projections. arXiv preprint arXiv:1811.10518.
14. Kelly Bickel, Pamela Gorkin, 2018. Numerical Range and Compressions of the Shift. arXiv preprint arXiv:1810.11680.
15. Hwa-Long Gau, Pei Yuan Wu, 2007. Numerical ranges of companion matrices. Linear algebra and its applications, 421, Elsevier, pp.202–218.
16. Ilya M Spitkovsky, Stephan Weis, 2018. Signatures of quantum phase transitions from the boundary of the numerical range. Journal of mathematical physics, 59, AIP Publishing, pp.121901.
numerical-range/properties.txt · Last modified: 2020/06/01 09:34 by plewandowska