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numerical-range:generalizations:q-numerical-range

# q-numerical range

## Definition

Let $A$ be an $N \times N$ matrix. The q-numerical range of $A$ is the set: $W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, |\braket{x}{y}| = q, \ket{x} \in \mathbb{C}^N, \ket{y} \in \mathbb{C}^N, \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}.$ Note that in the case $q=1$ we get back the standard numerical range.