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Properties of numerical range

Properties of numerical range $ W(A) $ of a matrix $ A $ of dimension $ N $ [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 = $ [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). In particular, we can consider elliptical shape (see [7]),
    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 $)
  8. if $ N=4 $:
    1. for \(A = \begin{pmatrix} \1 & C \\ D & \1 \end{pmatrix}\) the numerical range is the convex hull of two non-concentric ellipses [8],

More interesting properties you can find in [9], [10], [11], [12], [13], [14], [15], [16], [17]. In the case of tridiagonal matrices, the characteristic of the numerical range was presented in [18], [19]. For properties of the numerical range in the infinite dimension see [20], [21].

Application

An example application of numerical range can be found in [22].

Examples

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

Gau–Wu numbers

Definition

We define the Gau–Wu number of a matrix $ A $, denoted $ k(A) $, to be the maximum size of an orthonormal set $\{x_1,,x_k\} \subset \mathcal{C}^n$ such that the values $\bra{x_j} A \ket{x_j}$ lie on $ \partial W(A) $ [23].

The properties of Gau-Wu number was widely studied in [24], [25].

Rank-2 operators

Let $ W(A) $ will be the numerical range of matrix $A \in M_n(\mathrm{C})$ and let \begin{equation} \mathcal{F}_n = \{ W(A): A \in M_n(\mathrm{C}) \} \end{equation} be the set of all numerical ranges arising in $n$ dimensions.

Theorem

Let $ A $ be such operator that $ rank(A− \lambda \1)=2 $ for some $\lambda \in \mathbb{C}$ . Then $W(A)$ either is an element of $\mathcal{F}_3$, or the convex hull of two ellipses having a common focus, or has at most one flat portion on its boundary.

More properties (with examples) about rank-two operators is studied in [26].

References

  1. [1]R. Horn and C. Johnson, Topics in matrix analysis. Cambridge university press, 1994.
  2. [2]K. E. Gustafson and D. K. M. Rao, Numerical range: The Field of Values of Linear Operators and Matrices. Springer, 1997.
  3. [3]O. Toeplitz, “Das algebraische Analogon zu einem Satze von Fejer,” Mathematische Zeitschrift, vol. 2, no. 1, pp. 187–197, 1918, [Online]. Available at: https://link.springer.com/article/10.1007/BF01212904.
  4. [4]F. Hausdorff, “Der Wertevorrat einer Bilinearform,” Mathematische Zeitschrift, vol. 3, no. 1, pp. 314–316, 1919, [Online]. Available at: https://link.springer.com/article/10.1007/BF01292610.
  5. [5]F. D. Murnaghan, “On the field of values of a square matrix,” Proceedings of the National Academy of Sciences of the United States of America, vol. 18, no. 3, p. 246, 1932, [Online]. Available at: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076200/.
  6. [6]D. S. Keeler, L. Rodman, and I. M. Spitkovsky, “The numerical range of 3x3 matrices,” Linear Algebra and its Applications, vol. 252, no. 1-3, pp. 115–139, 1997, [Online]. Available at: https://dx.doi.org/10.1016/0024-3795(95)00674-5.
  7. [7]P. Rault, T. Sendova, and I. Spitkovsky, “3-by-3 matrices with elliptical numerical range revisited,” The Electronic Journal of Linear Algebra, vol. 26, 2013, [Online]. Available at: https://www.researchgate.net/publication/267480998_3-by-3_matrices_with_elliptical_numerical_range_revisited.
  8. [8]T. Geryba and I. M. Spitkovsky, “On some 4-by-4 matrices with bi-elliptical numerical ranges,” arXiv e-prints, vol. 0, 2020, [Online]. Available at: https://arxiv.org/abs/2009.00272.
  9. [9]M.-T. Chien, H. Nakazato, and J. Meng, “The diameter and width of numerical ranges,” Linear Algebra and its Applications, vol. 582, pp. 76–98, 2019, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379519303301.
  10. [10]J.-C. Bourin and A. Mhanna, “Positive block matrices and numerical ranges,” Comptes Rendus Mathematique, vol. 355, no. 10, pp. 1077–1081, 2017, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S1631073X1730256X.
  11. [11]N.-K. Tsin, “Diameter and minimal width of the numerical range,” Linear and multilinear algebra, vol. 14, no. 2, pp. 179–185, 1983, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081088308817554.
  12. [12]T. Geryba and I. M. Spitkovsky, “On the numerical range of some block matrices with scalar diagonal blocks,” Linear and Multilinear Algebra, vol. 0, pp. 1–14, 2020, [Online]. Available at: https://www.tandfonline.com/doi/abs/10.1080/03081087.2020.1749225?journalCode=glma20.
  13. [13]C.-K. Li and Y.-T. Poon, “Numerical Range Inclusion, Dilation, and Operator Systems,” arXiv preprint arXiv:1911.01221, vol. 0, 2019, [Online]. Available at: https://arxiv.org/abs/1911.01221.
  14. [14]M. Fatehi and A. Negahdari, “Numerical range of weighted composition operators which contain zero,” arXiv preprint arXiv:1901.07736, vol. 0, 2019, [Online]. Available at: https://arxiv.org/abs/1901.07736.
  15. [15]J. Kim and Y. Kim, “Jordan Plane and Numerical Range of Operators Involving Two Projections,” arXiv preprint arXiv:1811.10518, vol. 0, 2018, [Online]. Available at: https://arxiv.org/abs/1811.10518.
  16. [16]K. Bickel and P. Gorkin, “Numerical Range and Compressions of the Shift,” arXiv preprint arXiv:1810.11680, vol. 0, 2018, [Online]. Available at: https://arxiv.org/abs/1810.11680.
  17. [17]H.-L. Gau and P. Y. Wu, “Numerical ranges of companion matrices,” Linear algebra and its applications, vol. 421, no. 2-3, pp. 202–218, 2007, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379506001893.
  18. [18]I. Spitkovsky and C. Thomas, “Line segments on the boundary of the numerical ranges of some tridiagonal matrices,” The Electronic Journal of Linear Algebra, vol. 30, pp. 693–703, 2015, [Online]. Available at: https://doi.org/10.13001/1081-3810.3012.
  19. [19]R. T. Chien and I. M. Spitkovsky, “On the numerical ranges of some tridiagonal matrices,” Linear Algebra and its Applications, vol. 470, pp. 228–240, 2015, [Online]. Available at: https://doi.org/10.1016/j.laa.2014.08.010.
  20. [20]B. Lins and I. Spitkovsky, “Inverse continuity of the numerical range map for Hilbert space operators,” arXiv preprint arXiv:1810.04199, vol. 0, 2018, [Online]. Available at: https://arxiv.org/abs/1810.04199.
  21. [21]R. Birbonshi, I. M. Spitkovsky, and P. D. Srivastava, “A note on Anderson’s theorem in the infinite-dimensional setting,” Journal of Mathematical Analysis and Applications, vol. 461, no. 1, pp. 349–353, 2018, [Online]. Available at: https://doi.org/10.1016/j.jmaa.2018.01.002.
  22. [22]I. M. Spitkovsky and S. Weis, “Signatures of quantum phase transitions from the boundary of the numerical range,” Journal of mathematical physics, vol. 59, no. 12, p. 121901, 2018, [Online]. Available at: https://aip.scitation.org/doi/abs/10.1063/1.5017904.
  23. [23]H.-L. Gau, P. Y. Wu, and others, “Numerical ranges and compressions of Sn-matrices,” Operators and Matrices, vol. 7, no. 2, pp. 465–476, 2013, [Online]. Available at: https://www.researchgate.net/profile/Pei_Wu2/publication/268664189_Numerical_ranges_and_compressions_of_S_n_-matrices/links/547e5a590cf2d2200ede9933/Numerical-ranges-and-compressions-of-S-n-matrices.pdf.
  24. [24]K. A. Camenga, L. Deaett, P. X. Rault, T. Sendova, I. M. Spitkovsky, and R. B. J. Yates, “Singularities of base polynomials and Gau–Wu numbers,” Linear Algebra and its Applications, vol. 581, pp. 112–127, 2019, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379519302861.
  25. [25]K. A. Camenga, P. X. Rault, T. Sendova, and I. M. Spitkovsky, “On the Gau–Wu number for some classes of matrices,” Linear Algebra and its Applications, vol. 444, pp. 254–262, 2014, [Online]. Available at: https://www.sciencedirect.com/science/article/pii/S0024379513007830.
  26. [26]L. Rodman and I. M. Spitkovsky, “On numerical ranges of rank-two operators,” Integral Equations and Operator Theory, vol. 77, no. 3, pp. 441–448, 2013, [Online]. Available at: https://link.springer.com/article/10.1007/s00020-013-2092-y.