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Normalized numerical range


The classical numerical range $F(A)$ of $A\in M_n(\mathbb{C})$ is by definition the set of values of the corresponding quadratic form $x^*Ax$ on the unit sphere $S^n := \{x ^n : \|x \|=1 \}$of$^n$. Equivalently, \[ F(A) =\{(x^*Ax)/||x||2 xn\{0\}\}. \] Various modifications and generalization of the numerical range have been considered in the literature. Our paper is concerned with the so called normalized numerical range. Defined as \[ F_N(A) := \{ x ^n, Ax 0 \},\] it was introduced in [1], and then further investigated in for example [2].


Suppose that $A \in M_n(\mathbb{C})$. Then:

  • For all $z \in F_N(A)$, $|z| \le 1$.

*If $z \in F_N(A)$, then $|z| = 1$ if and only if $z = /||$for some$(A)$.

  • $F_N(A)$ is unitarily invariant: $F_N(U^*AU)=F_N(A)$ for any unitary $U\in M_n(\mathbb{C})$.

  • $F_N(e^{i\theta}A) = e^{i\theta} F_N(A)$ for all $ [0,2)$.

  • $F_N(cA) = F_N(A)$ for all $c > 0$.

  • If $A$ is invertible, then $F_N(A)$ is closed.

The following theorems and examples are taken from [3].

Theorem 1

Suppose that $A \in M_2(\mathbb{C})$ has non-zero eigenvalues $\lambda_1, \lambda_2$ such that $\lambda_1/\lambda_2 < 0$. Then $F_N(A)$ is a closed elliptical disk. In the case when $A > 0$, the ellipse is given by the equation

Theorem 2

For $A \in M_2 (\mathbb{C}) \backslash \{0\}$ with eigenvalues $\lambda_1$ and $\lambda_2$, the boundary of $F_N(A)$ is an ellipse if and only if $|\lambda_1| = |\lambda_2|$ or $\lambda_1/\lambda_2 < 0$.


1. The normalized numerical range of $A =


2. The normalized numerical range of $B =


3. The normalized numerical range of $C =


For matrix $A$ is a typical example of convex normalized numerical range that is not an ellipse. The normalized numerical range of matrix $B$ is not convex, but has a smooth boundary (boundary is differentiable). Finally, the last example for matrix $C$ has a boundary that is not smooth at one point. All three of these examples have boundaries that satisfy irreducible 8th degree polynomial equations.


  1. [1]W. Auzinger, “Sectorial operators and normalized numerical range,” Applied numerical mathematics, vol. 45, no. 4, pp. 367–388, 2003, [Online]. Available at:
  2. [2]L. Z. Gevorgyan, “Normalized numerical ranges of some operators,” Operators and Matrices, vol. 3, no. 1, pp. 145–153, 2009, [Online]. Available at:
  3. [3]B. Lins, I. M. Spitkovsky, and S. Zhong, “The normalized numerical range and the Davis–Wielandt shell,” Linear Algebra and its Applications, vol. 546, pp. 187–209, 2018, [Online]. Available at: