Numerical Shadow

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numerical-range [2013/04/16 22:23]
lpawela
numerical-range [2018/10/08 07:49] (current)
plewandowska
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 ===== Definition ===== ===== Definition =====
  
-For any square matrix $A$ of size $N$, one defines its //numerical range// (also called //field of values// [( :​murnaghan1932field )]) as a subset of the complex plane: ​+For any square matrix $A$ of dimension ​$d$, one defines its //numerical range// (also called //field of values// [( :​murnaghan1932field )]) as a subset of the complex plane: ​
 $$ $$
-W(A)=\{ z \in \mathbb{C}: z=\bra{\psi} A\ket{\psi},​ \ \ket{\psi} \in {\mathbb{C}}^N, \ \braket{\psi}{\psi}=1\}. ​+W(A)=\{ z \in \mathbb{C}: z=\bra{\psi} A\ket{\psi},​ \ \ket{\psi} \in {\mathbb{C}}^d, \ \braket{\psi}{\psi}=1\}. ​
 $$ $$
 Following a common convention we denote here the numerical range by $W(X)$, which goes back to the German term "​Wertevorrat"​ originally used by Hausdorff [( :​hausdorff1919wertvorrat )]. Following a common convention we denote here the numerical range by $W(X)$, which goes back to the German term "​Wertevorrat"​ originally used by Hausdorff [( :​hausdorff1919wertvorrat )].
numerical-range.1366151009.txt.gz · Last modified: 2013/04/16 22:23 by lpawela