The web resource on numerical range and numerical shadow

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numerical-range:properties

# 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 $d = \sqrt{\tr(AA^\dagger) - |\lambda_1|^2- |\lambda_2|^2}$ [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). 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 $$A = \left[ \begin{matrix} \alpha \1 & C\\ D & \beta \1 \\ \end{matrix} \right]$$ 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 $\{x_1,\ldots,x_k\} \subset \mathcal{C}^n$ 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 $A \in M_n(\mathbb{C})$ and let $$\mathcal{F}_n = \{ W(A): A \in M_n(\mathbb{C}) \}$$ 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. 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. Patrick Rault, Tsvetanka Sendova, Ilya Spitkovsky, 2013. 3-by-3 matrices with elliptical numerical range revisited. The Electronic Journal of Linear Algebra, 26.
8. Titas Geryba, Ilya M. Spitkovsky, 2020. On some 4-by-4 matrices with bi-elliptical numerical ranges. arXiv e-prints.
9. 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.
10. Jean-Christophe Bourin, Antoine Mhanna, 2017. Positive block matrices and numerical ranges. Comptes Rendus Mathematique, 355, Elsevier, pp.1077–1081.
11. Nam-Kiu Tsin, 1983. Diameter and minimal width of the numerical range. Linear and multilinear algebra, 14, Taylor and Francis, pp.179–185.
12. 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.
13. Chi-Kwong Li, Yiu-Tung Poon, 2019. Numerical Range Inclusion, Dilation, and Operator Systems. arXiv preprint arXiv:1911.01221.
14. Mahsa Fatehi, Asma Negahdari, 2019. Numerical range of weighted composition operators which contain zero. arXiv preprint arXiv:1901.07736.
15. Jaedeok Kim, Youngmi Kim, 2018. Jordan Plane and Numerical Range of Operators Involving Two Projections. arXiv preprint arXiv:1811.10518.
16. Kelly Bickel, Pamela Gorkin, 2018. Numerical Range and Compressions of the Shift. arXiv preprint arXiv:1810.11680.
17. Hwa-Long Gau, Pei Yuan Wu, 2007. Numerical ranges of companion matrices. Linear algebra and its applications, 421, Elsevier, pp.202–218.
18. Ilya Spitkovsky, Claire Thomas, 2015. Line segments on the boundary of the numerical ranges of some tridiagonal matrices. The Electronic Journal of Linear Algebra, 30, pp.693–703.
19. Ruey Ting Chien, Ilya M Spitkovsky, 2015. On the numerical ranges of some tridiagonal matrices. Linear Algebra and its Applications, 470, Elsevier, pp.228–240.
20. Brian Lins, Ilya Spitkovsky, 2018. Inverse continuity of the numerical range map for Hilbert space operators. arXiv preprint arXiv:1810.04199.
21. Riddhick Birbonshi, Ilya M Spitkovsky, PD Srivastava, 2018. A note on Anderson's theorem in the infinite-dimensional setting. Journal of Mathematical Analysis and Applications, 461, Elsevier, pp.349–353.
22. 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.
23. Hwa-Long Gau, Pei Yuan Wu, others, 2013. Numerical ranges and compressions of Sn-matrices. Operators and Matrices, 7, pp.465–476.
24. Kristin A Camenga, Louis Deaett, Patrick X Rault, Tsvetanka Sendova, Ilya M Spitkovsky, Rebekah B Johnson Yates, 2019. Singularities of base polynomials and Gau--Wu numbers. Linear Algebra and its Applications, 581, Elsevier, pp.112–127.
25. Kristin A Camenga, Patrick X Rault, Tsvetanka Sendova, Ilya M Spitkovsky, 2014. On the Gau--Wu number for some classes of matrices. Linear Algebra and its Applications, 444, Elsevier, pp.254–262.
26. Leiba Rodman, Ilya M Spitkovsky, 2013. On numerical ranges of rank-two operators. Integral Equations and Operator Theory, 77, Springer, pp.441–448.