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Numerical range


For any square matrix $A$ of dimension $d$, one defines its numerical range (also called field of values [1]) as a subset of the complex plane:

\[W(A)=\\{ z \in \mathbb{C}: z=\bra{\psi} A\ket{\psi}, \\ \ket{\psi} \in {\mathbb{C}}^d, \\ \braket{\psi}{\psi}=1\\}.\]

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 [2].

Properties and examples

See section Properties of numerical range.


  1. [1]F. D. Murnaghan, “On the field of values of a square matrix,” Proceedings of the National Academy of Sciences of the United States of America, vol. 18, no. 3, p. 246, 1932, [Online]. Available at:
  2. [2]F. Hausdorff, “Der Wertevorrat einer Bilinearform,” Mathematische Zeitschrift, vol. 3, no. 1, pp. 314–316, 1919, [Online]. Available at:

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