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# q-numerical range

## Definition

Let $A$ be an $d \times d$ matrix. The q-numerical range of $A$ is the set: $W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, \braket{x}{y} = q, \ket{x} \in \mathbb{C}^d, \ket{y} \in \mathbb{C}^d, \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}.$

The examples of application you can find in section applications-in-quantum-theory

## Properties

Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$ [1], [2],[3]:

1. Note that in the case $q=1$ we get back the standard numerical range;
2. $W_q(A)$ is a convex and bounded set (Tsing theorem) [4];
3. Unitarly invariant: $W_q(A) = W_q(UAU^\dagger)$ for any $U$ unitary matrix;
4. Transpose invariant: $W_q(A) = W_q(A^\top)$;
5. $W_{qz}(A) = zW_q(A)$ for any $z \in \mathbb{C}$, such that $|z|=1$;
6. For any $\mu, \eta \in \mathbb{C}$, we have $W_q(\mu A + \eta \1) = \mu W_q(A) + \eta q$.

## Example

The numerical aproximation of $W_{13/14}(A)$, where $$A = \begin{pmatrix} 0&1&1/2\\ 0&0&1\\ 0&0&0 \end{pmatrix}.$$

This example can be found in [5].

## Animation

To show the behavior of the $q$-numerical range for all $q \in [0,1]$ of the unitary matrix $U \in \mathcal{U}_3$ with eigenvalues $1, e^{ \frac{\pi \ii}{3}}$ and $e^{ \frac{2\pi\ii}{3}}$ you can see above animation.

1. Chi-Kwong Li, 1998. q-Numerical ranges of normal and convex matrices. Linear and Multilinear Algebra, 43, Taylor \& Francis, pp.377–384.
2. Mao-Ting Chien, Hiroshi Nakazato, 2006. The q-numerical range of a reducible matrix via a normal operator. Linear Algebra and its Applications, 419, North-Holland, pp.440–465.
3. Mao-Ting Chien, Hiroshi Nakazato, 2002. Davis--Wielandt shell and q-numerical range. Linear Algebra and its Applications, 340, Elsevier, pp.15–31.
4. Nam-Kiu Tsing, 1984. The constrained bilinear form and the C-numerical range. Linear Algebra and its Applications, 56, Elsevier, pp.195–206.
5. Ryuusuke Koide, Hiroshi Nakazato, 2008. The q-numerical range of a certain 3$\times$ 3 matrix. International Mathematical Forum3, Citeseer, pp.1001–1010.