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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 $A = \mathrm{diag}(1, 2, 4, 8)$.

numerical-range/generalizations/k-numerical-range.txt · Last modified: 2014/11/17 01:08 by lpawela