Numerical range
In the mathematical field of linear algebra and convex analysis, the numerical range or field of values or Wertvorrat or Wertevorrat of a complex n × n {\displaystyle n\times n} matrix A is the set W ( A ) = { x ∗ A x x ∗ x ∣ x ∈ C n , x ≠ 0 } = { ⟨ x , A x ⟩ ∣ x ∈ C n , ‖ x ‖ 2 = 1 } {\displaystyle W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ \mathbf {x} \neq 0\right\}=\left\{\langle \mathbf {x} ,A\mathbf {x} \rangle \mid \mathbf {x} \in \mathbb {C} ^{n},\ \|\mathbf {x} \|_{2}=1\right\}} where x ∗ {\displaystyle \mathbf {x} ^{*}} denotes the conjugate transpose of the vector x {\displaystyle \mathbf {x} } . The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing x equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing x equal to the eigenvectors).