# Numerical Shadow

The web resource on numerical range and numerical shadow

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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] Both sides previous revision Previous revision 2018/10/08 08:14 plewandowska [Definition] 2018/05/22 14:49 plewandowska [Definition] 2018/05/22 14:49 plewandowska [k-numerical range] 2018/05/22 14:04 plewandowska 2014/11/17 01:08 lpawela [Examples] 2014/11/17 01:05 lpawela 2014/11/16 23:46 lpawela [Definition] 2014/11/16 23:45 lpawela [Definition] 2014/10/15 00:36 lpawela 2014/10/15 00:33 lpawela created 2018/10/08 08:14 plewandowska [Definition] 2018/05/22 14:49 plewandowska [Definition] 2018/05/22 14:49 plewandowska [k-numerical range] 2018/05/22 14:04 plewandowska 2014/11/17 01:08 lpawela [Examples] 2014/11/17 01:05 lpawela 2014/11/16 23:46 lpawela [Definition] 2014/11/16 23:45 lpawela [Definition] 2014/10/15 00:36 lpawela 2014/10/15 00:33 lpawela created Line 3: Line 3: - 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 + Let $A$ be an $d \times ​d$ 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\} Line 10: Line 10: 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 