Numerical Shadow

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numerical-range

Numerical range

Definition

For any square matrix $A$ of size $N$, one defines its numerical range (also called field of values [1]) as a subset of the complex plane: $$ W(A)=\{ z \in \mathbb{C}: z=\bra{\psi} A\ket{\psi}, \ \ket{\psi} \in {\mathbb{C}}^N, \ \braket{\psi}{\psi}=1\}. $$ 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 [2].

Properties and examples

References

1. Murnaghan, F. D, 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, National Academy of Sciences, pp.246.
2. Hausdorff, F., 1919. Der Wertevorrat einer Bilinearform. Mathematische Zeitschrift, 3, Springer, pp.314–316.
numerical-range.txt · Last modified: 2013/04/16 22:23 by lpawela