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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 $$W_c(A) = \left\{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \right\}.$$ Let $C=\mathrm{\;diag}(c_1, c_2, \ldots, c_d)$.
Then $\mu \in W_c(A) \iff \mu = \tr(CU^\dagger AU)$ for some unitary matrix $U$. This fact motivates to define $C$-numerical range by $$W_C(A) = \left\{ \tr(CU^\dagger AU): \mathrm{ \;U \; unitary\; }\right\}$$ for any square matrix $C$. For convenience one may define the unitary similarity orbit of matrix $A$ given by the formula $$\mathcal{U}(A) = \left\{U^\dagger AU :\mathrm{\;U \; unitary\; }\right\}.$$

## Properties

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

1. $W_C(A)$ is a compact set;
2. Symmetry: $W_C(A) = W_A(C)$;
3. For any matrices $A, B$ of dimension $d \times d$, $W_C(A+B) \subseteq W_C(A) + W_C(B)$;
4. For any $\mu, \eta \in \mathbb{C}$, then $W_C(\mu A + \eta \1) = \mu W_C(A) + \eta \tr(C)$;
5. $W_C(U^\dagger AU) = W_C(A)$ for any unitary matrix $U$;
6. $W_{V^\dagger CV}(A) = W_C(A)$ for any unitary matrix $V$;
7. $W_C(A) = W_C(A^\top) \iff C^\top \in \mathcal{U}(C)$
8. $\overline{W_C(A)} = W_C(\overline{A}) \iff \overline{C} \in \mathcal{U}(C)$
9. $\overline{W_C(A)} = W_C(A^\dagger) \iff C^\dagger \in \mathcal{U}(C)$
10. If $c=[1,0,\ldots,0]$, then $W_c(A)$ reduces to the classical numerical range of $A$ and if $C=\mathrm{\;diag}(1,0,\ldots,0)$, then $W_C(A)$ reduces to the classical numerical range of $A$.

## Convexity

Convexity of $W_c(A)$ and $W_C(A)$ of a matrix $A$ of dimension $d \times d$:

1. $W_c(A)$ is a convex set (Westwick theorem): [1]
2. $W_C(A)$ is a convex set if one of the following holds: [2]
• there exist $\mu$, $\eta \in \mathbb{C}$ with $\mu \neq 0$ such that $\mu C + \eta \1$ is hermitian;
• there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ is unitarity similar to $(C_{ij})_{1 \le i,j \le d}$ in block form, where $C_{ii}$ are square matrices and $C_{ij} = 0$ if $i \neq j+1$;
• there exist $\mu \in \mathbb{C}$ such that $C - \mu \1$ has rank one.
1. R Westwick, 1975. A theorem on numerical range. Linear and Multilinear Algebra, 2, Taylor & Francis, pp.311–315.
2. Chi-Kwong Li, 1994. C-numerical ranges and C-numerical radii. Linear and Multilinear Algebra, 37, Taylor & Francis, pp.51–82.