# Numerical range

In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n × n matrix A is the set

$W(A)=\left\{{\frac {\mathbf {x} ^{*}A\mathbf {x} }{\mathbf {x} ^{*}\mathbf {x} }}\mid \mathbf {x} \in \mathbb {C} ^{n},\ x\not =0\right\}$ where x* denotes the conjugate transpose of the vector x.

In engineering, numerical ranges are used as a rough estimate of eigenvalues of A. Recently, generalizations of numerical range are used to study quantum computing.

A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e.

$r(A)=\sup\{|\lambda |:\lambda \in W(A)\}=\sup _{\|x\|=1}|\langle Ax,x\rangle |.$ ## Properties

1. The numerical range is the range of the Rayleigh quotient.
2. (Hausdorff–Toeplitz theorem) The numerical range is convex and compact.
3. $W(\alpha A+\beta I)=\alpha W(A)+\{\beta \}$ for all square matrix A and complex numbers α and β. Here I is the identity matrix.
4. $W(A)$ is a subset of the closed right half-plane if and only if $A+A^{*}$ is positive semidefinite.
5. The numerical range $W(\cdot )$ is the only function on the set of square matrices that satisfies (2), (3) and (4).
6. (Sub-additive) $W(A+B)\subseteq W(A)+W(B)$ .
7. $W(A)$ contains all the eigenvalues of A.
8. The numerical range of a 2×2 matrix is an elliptical disk.
9. $W(A)$ is a real line segment [α, β] if and only if A is a Hermitian matrix with its smallest and the largest eigenvalues being α and β
10. If A is a normal matrix then $W(A)$ is the convex hull of its eigenvalues.
11. If α is a sharp point on the boundary of $W(A)$ , then α is a normal eigenvalue of A.
12. $r(\cdot )$ is a norm on the space of n×n matrices.
13. $r(A)\leq \|A\|\leq 2r(A)$ , where $\|\cdot \|$ denotes the operator norm.
14. $r(A^{n})\leq r(A)^{n}$ 