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numerical-range:properties

# Properties of numerical range

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

1. $W(A)$ is a compact subset of $\mathbb{C}$,
2. $W(A)$ is convex set (Hausdorf-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 $d = \sqrt{AA^\dagger - |\lambda_1|^2- |\lambda_2|^2}$ [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 $A$)

## Examples

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

### 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.