The web resource on numerical range and numerical shadow

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].

## 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.