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numerical-range:generalizations:k-numerical-range

# k-numerical range

## Definition

Let $A$ be an $n \times n$ matrix and $P_k$ be a projector of rank $k$. The k-numerical range of $A$ is the set $W_k(A) = \left\{ z \in \mathbb{C}: z=\frac{1}{k}\Tr P_k A \right\}$

Note that, this numerical range is different from the higher-rank-numerical-range as for a Hermitian matrix $A$, we get $W_k = \left[\frac{1}{k}\sum_{i=1}^k\lambda_i, \frac{1}{k}\sum_{i=0}^{k-1} \lambda_{N-i} \right].$ where $\lambda_i$ are the eigenvalues of $A$ in an increasing order. On the other hand, the higher-rank-numerical-range is given by $\Lambda_k(A) = [\lambda_k, \lambda_{N-k+1}],$ Hence, we get $\Lambda_k(A) \subset W_k(A).$ We should note that in the case when $k=1$ the k-numerical range becomes the standard numerical range $W_1(A) = W(A) = \Lambda_1(A).$

## Examples

A comparison between the k-numerical range and higher-rank numerical range in the case $k=2$. Note that $\Lambda_2 \subset W_2$. The matrix used in this example is $A = \mathrm{diag}(1, 2, 4, 8)$.