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numerical-range:generalizations:higher-order-numerical-range
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numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:45] lpawela |
numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:56] (current) lpawela |
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| $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections | $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections | ||
| of conveniently computable half--planes in $\mathbb{C}$. | of conveniently computable half--planes in $\mathbb{C}$. | ||
| - | Among the many more recent papers concerning the $\Lambda_k(M)$, let us mention CITE!! [LPS2009,GLW2010]. | + | Among the many more recent papers concerning the $\Lambda_k(M)$, let us mention [( :li2009condition )] and [( :gau2010higher )]. |
| - | Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and \.Zyczkowski (see [CK\.Z2006a,CK\.Z2006b]) | + | Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and Życzkowski (see [( :choi2006higher )]) |
| defined the rank--$k$ numerical range of $M$ as | defined the rank--$k$ numerical range of $M$ as | ||
| \[ | \[ | ||
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| where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard | where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard | ||
| to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that | to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that | ||
| - | there is some $k$--dimensional subspace $S$ of $\mathbb{C}^N$ such that $(Mu,u)=\lambda$ for \textbf{all} | + | there is some $k$--dimensional subspace $S$ of $\mathbb{C}^N$ such that $(Mu,u)=\lambda$ for **all** |
| unit vectors in $S$. In particular, we see that | unit vectors in $S$. In particular, we see that | ||
| \[ | \[ | ||
| W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad. | W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad. | ||
| \] | \] | ||
numerical-range/generalizations/higher-order-numerical-range.txt · Last modified: 2013/05/08 12:56 by lpawela
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