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numerical-range:generalizations:k-numerical-range

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numerical-range:generalizations:k-numerical-range [2018/05/22 14:49]
plewandowska [Definition]
numerical-range:generalizations:k-numerical-range [2018/10/08 08:14] (current)
plewandowska [Definition]
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-Let $A$ be an $\times ​n$ matrix and $P_k$ be a projector of rank $k$. The $k$-numerical range of $A$ is the set +Let $A$ be an $\times ​d$ 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\} W_k(A) = \left\{ z \in \mathbb{C}: z=\frac{1}{k}\Tr P_k A \right\}
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 Note that, this numerical range is different from the [[numerical-range:​generalizations:​higher-rank-numerical-range | higher-rank-numerical-range]] as for a Hermitian matrix $A$, we get Note that, this numerical range is different from the [[numerical-range:​generalizations:​higher-rank-numerical-range | higher-rank-numerical-range]] as for a Hermitian matrix $A$, we get
 \[ \[
-W_k = \left[\frac{1}{k}\sum_{i=1}^k\lambda_i,​ \frac{1}{k}\sum_{i=0}^{k-1} \lambda_{N-i} \right].+W_k = \left[\frac{1}{k}\sum_{i=1}^k\lambda_i,​ \frac{1}{k}\sum_{i=0}^{k-1} \lambda_{d-i} \right].
 \] \]
 where $\lambda_i$ are the eigenvalues of $A$ in an increasing order. On the other hand, the [[numerical-range:​generalizations:​higher-rank-numerical-range | higher-rank-numerical-range]] is given by where $\lambda_i$ are the eigenvalues of $A$ in an increasing order. On the other hand, the [[numerical-range:​generalizations:​higher-rank-numerical-range | higher-rank-numerical-range]] is given by
 \[ \[
-\Lambda_k(A) = [\lambda_k, \lambda_{N-k+1}],+\Lambda_k(A) = [\lambda_k, \lambda_{d-k+1}],
 \] \]
 Hence, we get Hence, we get
numerical-range/generalizations/k-numerical-range.txt · Last modified: 2018/10/08 08:14 by plewandowska