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numerical-range:generalizations:c-numerical-range

# $C$-numerical range

## Definition

Let $A$ be an $d \times d$ matrix and $(c_1, c_2, \ldots, c_d)$ be a real d-tuple. The $C$-numerical range of $A$ is the set $$W_C(A) = \left\{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \right\}$$

## Properties

Properties of $W_C(A)$ of a matrix $A$ of dimension $d \times d$:

1. $W_C(A)$ is a convex set (Westwick theorem) [1];
2. If $c=[1,0,\ldots,0]$, then $W_C(A)$ reduces to the classical numerical range of $A$.
1. R Westwick, 1975. A theorem on numerical range. Linear and Multilinear Algebra, 2, Taylor \& Francis, pp.311–315.